Can a person sit Shiva for a missing person-Mourning if there is no identified body of the deceased:
Deceased was a married male: In all cases that the body is missing or unidentifiable, and death cannot be determined to the point that the wife of the deceased can remarry [as explained in Even Haezer chapter 17], then the relatives do not sit Shiva or Shloshim. [If the death has been Halachically determined, the relatives may sit Shiva even if knowledge of this has not yet reached the wife.]
Deceased was not married or is a female: If the deceased was not married, then some Poskim rule that one begins sitting Shiva and Shloshim as soon as word spreads of the persons death [or if majority of people die as a result of whatever occurrence happened to the deceased], and they give up hope of retrieving the body for burial. Other Poskim, however, rule that one always requires the same level of proof as required to allow a wife to remarry when a married man passes away, in order to sit Shiva/Shloshim. [This dispute would seemingly likewise apply if a woman passes away, whether married or single.] Practically, the relatives may choose to be stringent to mourn the deceased even if sufficient proof has not been brought to allow a wife to remarry.
Sons saying Kaddish in above cases: Some Poskim rule that whenever Shiva/Shloshim is not kept due to insufficient proof of death, then likewise Kaddish is not recited by the sons. However, the sons may go up for Maftir and Daven for the Amud on occasion, and should do other matters for the merit of his soul such as giving charity. Other Poskim, however, rule that the sons may say Kaddish.
Mourning if person was a Goses: A person in a state of Goses is assumed to not be able to live for more than three days, and hence he can be assumed to be dead after the passing of three days, and his relatives are to mourn him, if they are out of town and have no way of verifying whether he is still alive. See Chapter 2 Halacha 1A for the full details of this subject!
Testimony which obligates keeping Aveilus: In order for one to be obligated to mourn the passing of a relative, one must receive authentic information regarding his death. A single witness is believed in this regard, and one is not required to hear testimony from two witnesses. Likewise, a person is believed even if he did not personally witness the death, but is carrying the information from someone who did witness it [i.e. another relative, the doctor, hospital etc]. [Likewise, a woman is believed.] Even a gentile’s testimony is believed if he is Mesiach Lefi Tumo [i.e. a casual statement without any incentive for the gentile]. [The general rule is as follows: All testimony that is valid to allow a woman to remarry, is valid for determining the death of the relative to obligate Aveilus. Thus, the following forms of testimony are all valid: Slave, woman, relative, and Eid Mepi Eid. One who is an invalid witness due to sin, is only valid if he is Mesiach Lefi Tumo. A child, however, is not believed, and neither is one who is insane.]
Contradiction of testimony: In the event that there is conflicting testimony regarding the death of the relative, with two people saying that he died and two others saying that he did not die, then one does not mourn him.
One only begins to sit Shiva for a missing person if:
1. There is testimony that he passed away on the level that it would allow a wife to remarry.
2. The person is missing, and rumor has it that he has deceased, and one has given up hope of finding the body for burial, and the deceased person is not a married male.
 Shach 375:7 and Taz 375:3, in implication of Michaber ibid who writes “Mayim Sheyeish Lahem Sof”; Rama E.H. 17:5; Beis Yosef 375 in name of Ittur in name of Teshuvos Harif; Gilyon Maharsha 375
Other opinions: Some Poskim rule that so long as the word of if death has spread, and the relatives give up hope of finding the body, they begin Shiva even though the wife may not yet Halachically remarry. [Rav Haiy and Sherira Gaon, brought in Tur 375 and Shach ibid; Mishkanos Yaakov 69, brought in Pischeiy Teshuvah 375:3, that so rule majority of Rishonim such as Raavan, Maharam, Rashba, Rosh and Tur, and so is implied from Michaber 339:2]
 The reason: As allowing them to mourn can cause a great mishap to occur, as people who saw the relatives in mourning will testify that the woman’s husband passed away and allow her to remarry even though in truth there was not sufficient Halachic proof to allow her to remarry, and she is thus stumbling on the prohibition of Eishes Ish. [Shach ibid; Poskim ibid]
 See Shvus Yaakov 2:114; Pischeiy Teshuvah 375:3
 See Chasam Sofer 344, brought in Pischeiy Teshuvah 375:3; See Ezras Nashim E.H. 17:2, brought in Rebbe Akiva Eiger 375:7 that perhaps we apply a Lo Plug to even one who is not married, and that the level of proof required to allow a wife to remarry is required here as well, or if we do not extend it to where one is single and hence, one who is not married is to be mourned as soon as the rumor of his death has spread.
 Shvus Yaakov 1:103; Pischeiy Teshuvah ibid
 See Mishkanos Yaakov 69, brought in Pischeiy Teshuvah 375:3
 Implication of Shach 375:7 [so writes Chasam Sofer ibid]; Conclusion of Ezras Nashim E.H. 17:2 [of Maharam Chaviv]
 Chasam Sofer ibid “One who chooses to mourn, may do so and is blessed, as he relies on the ruling of the Shvus Yaakov”
 Pischeiy Teshuvah 375:3
 Chinuch Beis Yosef 91, brought in Pischeiy Teshuvah ibid; Gilyon Maharsha 375
 The reason: As allowing them to say Kaddish can cause a great mishap to occur, as people who see this will testify that the woman’s husband passed away and allow her to remarry even though in truth there was not sufficient Halachic proof to allow her to remarry, and she is thus stumbling on the prohibition of Eishes Ish. [ibid]
 Poskim ibid
 Dissenting opinion in Chinuch Beis Yosef ibid; Mishkanos Yaakov 69, brought in Pischeiy Teshuvah 375:3, rules that in truth most Poskim rule that even Aveilus may be kept, and certainly Kaddish may be said; Kneses Yechezkal 53 permits them to say Kaddish in the house of the wife.
 Michaber 397:1
 See Taz 397:1; Michaber E.H. 17:3
 Michaber ibid; Rashal in name of Ravayah, brought in Taz 397:1; See Michaber Even Haezer 17:3; 14-16
Other opinions: Some Poskim rule a gentile is not believed to testify that someone passed away regarding keeping the laws of Aveilus. [Taz 397:1 in name of Rashal, in name of Or Zarua]
 Shach 397:1 in name of Ramban and Bach; See Even Haezer 17
 Michaber Even Haezer 17:3
 Michaber Even Haezer 17:13
 Michaber 397:2
 The reason: As Safek Aveilus Lihakel. [Taz 397:2] Alternatively, as we establish a person on his Chazaka that he is alive. [Shach 397:2]