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Saying Shehechiyanu on the second day:
The blessing of Shehechiyanu is recited on both the first and second day of Rosh Hashanah. It is said by [candle lighting or by] Kiddush and the next day by Shofar. Nevertheless initially when saying the blessing of Shehechiyanu on the second night [by candle lighting or] Kiddush, and by the Shofar blowing the next day, one is to wear a new garment that requires Shehechiyanu to be recited while wearing it. Alternatively for Kiddush one may take a new fruit and place it in front of him before Kiddush. Whatever one chooses to use, upon reciting Shehechiyanu he is to intend to include within the blessing also the new fruit or garment. [However one is not to intend only on the new fruit or garment and is to mainly intend that the Shehechiyanu is being said over the Yom Tov.] If one does not have a new fruit or new garment available, one is nevertheless to say the blessing of Shehechiyanu.
Shehechiyanu by candle lighting: See Halacha 25A for the full details of this subject!
Shehechiyanu by Kiddush: See Chapter 2 Halacha 4 for the full details of this subject!
Shehechiyanu by Shofar: See “Shofar” Chapter 2 Halacha 9B for the full details of this subject!
On which clothing is the blessing of Shehechiyanu to be recited over?
The blessing of Shehechiyanu may only be recited on clothing of importance of which one rejoices over its purchase [such as a new Kapata; Tallis Gadol; suit]. However an item which is not of importance one may not say Shehechiyanu over it. Thus the blessing may not be said over new shirts, pants, shoes or undergarments. The custom in these provinces is not to say a blessing over these clothing even if one is a pauper and hence rejoices over purchasing them. [The above law applies throughout the year. Nevertheless regarding the Shehechiyanu of the 2nd day of R”H many are lenient to use a new clothing, such as a shirt or Yarmulke and the like, that one normally does not recite Shehechiyanu on during the year. Some Poskim however rule that one is to wear an important clothing that receives Shehechiyanu even during the year. The Rebbe was seen to use a new scarf or handkerchief as the new clothing for the second day of R”H.]
When is the blessing of Shehechiyanu recited upon buying important clothing?
It is to be recited immediately after the acquisition is completed, which is the time of his joy, so long as the clothing is ready to be worn without further mending. If it does require further mending then the blessing is to be said immediately after wearing the clothing. In all cases, if one did not say the blessing at the time of the purchase, it is to be said immediately upon wearing the clothing, or any time within the first wearing. [Accordingly one should not buy a new clothing and then delay the Shehechiyanu until the 2nd day of R”H unless the clothing required mending.]
May the blessing still be said if one already wore it one time? 
The blessing may not be said by the second time one is wearing the clothing. If one removed it after the first wearing with intent to re-wear right away, [and he went ahead and re-wore it right away] then the blessing may still be said.
Can one use the new fruit for Shofar or candle lighting?
Some Poskim rule a new fruit cannot be used for Shofar or candle lighting as one will not be eating it in close proximity to the blessing of Shehechiyanu. However some Poskim mention the option of using a new fruit even by candle lighting [however not by Shofar]. However see next Halacha 25A that some of these Poskim rule that in such a case the candles must be lit in close proximity to the meal!
 600/5-6; Siddur Admur; Tur 600; Michaber 600/2-3
Background from Admur:
*For a clear understanding of some of the concepts and matters below refer to chapter 3 where the background of the two days of Yom Tov is explained in great length and detail.
First opinion: By a regular Yom Tov in the Diaspora they recite Shehechiyanu in Kiddush on both days. The reason for this is because the entire reason of keeping two days in the Diaspora is because of the doubt of their ancestors as to (on which day the month was sanctified and consequently as to) which day is Yom Tov. Now the ancestors always said Shehechiyanu also on the second day as perhaps the second day is Yom Tov and the previous day was a weekday and hence it was not exempt with the blessing of Shehechiyanu said yesterday. However on the second night of R”H there are opinions [Geonim; brought in Tur ibid] that say one does not recite Shehechiyanu. Their reason is because the two days of Rosh Hashanah are considered one level of Holiness and like one long day, and thus the second day was included within the blessing of Shehechiyanu recited on the 1st night, and hence why should he repeat the blessing a second time. They apply the same ruling towards the Shehechiyanu of Shofar on the second day that one is not to say Shehechiyanu over the Shofar on the second day as it was already included within the Shehechiyanu of the 1st day.
Second opinion: Others [Rashi; Rashbam; Rosh; brought in Tur] however rule that one is required to recite a second blessing of Shehechiyanu on the second day both by Kiddush and blowing Shofar. The reason for this is because the entire reason that the two days are considered like one long day is due to the custom of our ancestors, and even our ancestors repeated the Shehechiyanu on the second day. The reason is as follows: In those times the month was sanctified based on testimony and they thus considered both days as one long day as they suspected that perhaps the witnesses came after Mincha and hence the two days are considered like one Holiday and one set of Holiness. Nevertheless our forefathers would recite the blessing of Shehechiyanu also on the second day, despite the fact that it was all one long stretch of Holiness and one long day. The reason they did so is because even if witnesses were to arrive after Mincha [and hence cause both days to be one long day] nevertheless the main Biblical day of Yom Tov would be on the 2nd day while the 1st day would be considered a Rabbinical Yom Tov and is considered like a weekday in comparison to the second day which was set to be the first of the month. Therefore since even our ancestors recited Shehechiyanu on the 2nd day also we should recite Shehechiyanu on the second day (even in Eretz Yisrael) just like is done by every Holiday in order not to swerve from the custom of our ancestors. [ibid; In summary: Since even in the times of our ancestors Shehechiyanu was said on both days, being the second day was the Biblical holiday and thus could not be exempt with the blessing said on the first day, therefore we too make a separate blessing on each day, as the entire concept of keeping two days is so we follow the same pattern as our ancestors.] [600/5]
The final ruling: Practically we rule like the second opinion which holds that Shehechiyanu is said also on the second day [by both Kiddush and Shofar]. Nevertheless initially it is good to avoid the dispute regarding the blessing and hence wear a new garment that requires Shehechiyanu [Admur 600/6; Rama 600/2] or eat a new fruit and have them in mind when reciting the blessing. [600/6; Siddur Admur [“so is the custom”]; Michaber 600/2; Rosh and Maharam Meir of Rothenberg brought in Tur] Nevertheless if one does not have a new clothing and new fruit then nevertheless he is to recite Shehechiyanu as the main opinion follows the second opinion [Admur ibid; Siddur Admur; Michaber ibid; Taz 600/1]
Ruling of Admur in Siddur: In 600/6 Admur writes it is proper to place a new fruit. In the Siddur he writes it is the custom to place a new fruit. Piskeiy Hassidur  explains that Admur in the Siddur is following the opinion of the Shaar Hakavanos and Mishnas Chassidim which rules there is no need to have a new fruit placed. He hence writes that it is merely the custom to do so and not that one should do so.
 Second opinion in Admur ibid and final ruling of Admur; Michaber 600/2 [regarding Kiddush];
 Admur 600/6 [Kiddush] 600/7 [Shofar]; Rama 600/3 based on Hagahos Maimanis; Darkei Moshe 600
Michaber and custom of Sefaradim: The Michaber rules there is a difference between the Shehechiyanu of Kiddush and Shofar on the second day. Regarding Kiddush one says Shehechiyanu also on the 2nd day [600/2] However regarding Shofar one does not repeat Shehechiyanu by Shofar of the second day. [600/3; Beis Yosef 600] Practically this is the custom of Sefaradic Jewry, not to say Shehechiyanu by Shofar of the second day. [Admur 600/7; Kaf Hachaim 600/3]
 Elya Raba 600/3; Shaareiy Teshuvah 600/1; Mateh Efraim 581/54; Alef Hamagen 581/126; Kaf Hachaim 600/6; Hiskashrus 898; As for why the placing of a new fruit or garment by candle lighting is not mentioned by earlier Poskim see “The Laws of Erev R”H” Halacha 18 in footnotes that the saying of Shehechiyanu by candle lighting is not mentioned in previous Poskim.
 Admur 600/6 [Kiddush]; 600/7 [Shofar]; Rama 600/2 [Kiddush; however by Shofar in 600/3 having a new garment is not mentioned-see Kaf Hachaim 600/10]; M”A 600/3 [Shofar]; Hagahos Maimanis; Elya Raba 600/3; M”B 600/3; The option of new clothing is omitted from Siddur regarding Kiddush
Other Opinions: Some rule one is not required to search for a new fruit or garment and Shehechiyanu may be recited even initially. [Birkeiy Yosef 600/2; Maaseh Rav 203; Shaar Hakavanos p. 90; Mishnes Chassidim; brought in Piskeiy Hassidur 178; Kaf Hachaim 600/7; See Kaf Hachaim 600/10 regarding Shofar; However see Peri Eitz Chaim that one is to search for a new fruit. The Piskeiy Hasiddur says this is a mistake in the wording. The Kaf Hachaim ibid quotes the Peri Eitz Chaim 25/7 that one is not required to search for a new fruit.] The Piskeiy Hassidur  explains that Admur in the Siddur holds like the opinion of the Shaar Hakavanos and Mishnas Chassidim and he thus wrote that it is merely the custom to do so and not that one should do so.
What clothing today require Shehechiyanu? See Q&A!
Other Yomim Tovim: It is clearly evident that there is no need to have a new fruit on the second day of other Holidays, and those that are accustomed to do so are mistaken. [M”A 600/2; Elya Raba 600/4; M”B 600/5; Kaf Hachaim 600/8]
 Admur 600/6; Michaber ibid
Can one use the new fruit for Shofar or candle lighting? Regarding Kiddush Admur ibid gives both options, however regarding Shofar he only mentions the new clothing. See Q&A for the details of this matter!
 Admur ibid
 M”E 600/13; Minchas Shlomo 20; Piskeiy Teshuvos 600/2
 Admur 600/6 [Kiddush]; 600/7 [Shofar-only Lechatchila]; Michaber 600/2 regarding Kiddush
 Seder 12/5; See Mateh Efraim 600/11
 So is the custom I witnessed amongst a number of Baalei Tokeia and Moreh Horaas and so was the custom of the Rebbe as brought next.
Source for this custom: The Poskim rule that one may say the Shehechiyanu on the 2nd night over new grape juice even if one already ate grapes during the year [Shvus Yaakov 3 brought in Shaareiy Teshuvah 600/1; Kaf Hachaim 600/5; Alef Hamagen 600/8 ] The reason for this is because most Poskim rule one is in any event to say Shehechiyanu on the 2nd night and hence although during the year we don’t say Shehechiyanu over new grape juice, if one already ate grapes during the year, nevertheless in this case one can be lenient. [ibid] Thus we can apply the same logic here that although we don’t normally say Shehechiyanu on non-important clothing during the year, nevertheless on R”H one can be lenient to do so.
 Mateh Efraim 600/11; see Mateh Efraim 600/9 that rules one is not to use new grape juice if he already ate new grapes and is hence to take a different fruit.
 Otzer Minhagei Chabad 245 regarding the custom of the Rebbe Rashab ; Otzer 247 regarding the Rebbe’s custom
 Seder 12/4; 223/4; M”A 600/1; Machatzis Hashekel ibid; M”B 600/3; Kaf Hachaim 600/4; Ketzos Hashulchan 64/2; See Mateh Efraim 600/11
 M”E 600/11-12; Machatzis Hashekel 600/1; See previous Q&A that many are lenient to wear a clothing that is not important enough for Shehechiyanu during the year but suffices for the 2nd night of R”H to avoid the Safek.
 Seder 12/4; Ketzos Hashulchan 64/2
 Machatzis Hashekel 600/1 based on M”A 600/3 [regarding Shofar]; Admur 600/7 [regarding Shofar] M”E 581/54 [regarding candle lighting] only mention the option of a new clothing.
 Thus not only does it not help avoid the Safek Bracha of Shehechiyanu of the 2nd night, it even enters one to a new Safek which is must he now still say a Shehechiyanu over this fruit. This is dependent on if he truly had joy upon seeing the fruit during the blessing. [Machatzis Hashekel]
Why can’t the Shehechiyanu count for the seeing of the fruit? Ideally the blessing of Shehechiyanu over a fruit is to be said when one sees the fruit and not upon eating it. [See 225/3; Admur in Seder 11/12]. However we are not accustomed to do so during the year even if one has joy from seeing the fruit [ibid]. Looking at the fruit not suffice to remove oneself from the doubt of the 2nd day of R”H, as seemingly one has already seen the new fruit many times prior [when buying it etc] and the blessing should have already been said then, and whether a blessing may be said upon the second time of seeing it is itself a dispute. [Machatzis Hashekel ibid]
 Elya Raba 600/3; Shaareiy Teshuvah 600/1; Mateh Efraim 581/54; Alef Hamagen 581/126; Kaf Hachaim 600/6; Hiskashrus 898
The reason: As in truth the blessing of Shehechiyanu over a fruit is to be said when one sees the fruit and not upon eating it. [See 225/3; Admur in Seder 11/12]. Now although we don’t rule this way during the year unless one has joy from seeing the fruit [ibid] nevertheless it suffices to remove oneself from the doubt of the 2nd day of R”H. [Kaf Hachaim ibid] Vetzaruch Iyun as by doing so one enters into another doubt regarding if he must repeat Shehechiyanu later on, when eating the fruit, as writes the Machatzis Hashekel ibid.