May one wear slippers for Davening?
One is required to Daven in a dress code that is fitting to be worn before a king, aristocrat, and people of stature. From this it is understood that whether it is permitted to Daven wearing slippers is dependent on the custom of one’s society, and as to whether it is acceptable to wear them when meeting with a person of stature and honor. Accordingly, if one would feel comfortable wearing his slippers when greeting honorable guests, then he may likewise wear them while Davening. If, however, one would not feel comfortable doing so, then it should not be worn. The above limitation only applies to the time of Shemoneh Esrei, however, prior to, and post, Shemoneh Esrei, one may Daven with slippers. [All the above is from the letter of the law, however, it is proper for one to do his utmost to be dressed honorably for Davening and hence it is best for one to wear shoes throughout Davening.]
One may be lenient to wear slippers during Davening, if they are clean and respectable to receive guests in them. Nonetheless, it is best to wear proper shoes for Davening, and if the slippers are not acceptable to be worn before respectable people, then from the letter of the law, they may not be worn for Shemoneh Esrei.
 See Mishnas Yosef 4:4; Leket Hakemach Hachadash 91:9; Shevet Hakehasi 3:41; Orchos Chaim [Spinka] 91:3; Ishei Yisrael 10:6; Piskeiy Teshuvos 91:4
 See Admur 90:1 [i.e. there is a dress code for Davening]; Admur 91:5, Michaber 91:5, Rambam Tefila 5:5 [Whether one may be barefoot follows the community custom of whether it is accepted to stand barefoot before the Gedolim, and the same would apply to the general dress code]; See Admur 74:7 based on Rashi Brachos 25b regarding the requirement to Daven while wearing a shirt that “by Tefila of Shemoneh Esrei one needs to view himself as if he is standing before the king”
 Orchos Chaim ibid
 Piskeiy Teshuvos ibid; So is the custom of many on Yom Kippur; See Admur 91:5 and Michaber 91:5 who negate Davening barefoot in certain areas that are accustomed to stand before Gedolim with Batei Raglayim, which would imply that to Daven with socks on is permissible, even if one is not wearing any shoes.
 See Admur 74:9
 See Rioshumei Aaron 1:19 that Rav Moshe Feinstein would remove his slippers and wear shoes for Davening, even when Davening at home.