IslamQA
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Summary (from the source)
It is permissible for a man to pray in a vest or undershirt if it is clean and covers the `Awrah. Men only have to cover the area between the navel and the knee when praying.
Question
Is it permissible to pray in an undershirt, i.e., a vest [sleeveless undershirt], at home?
Answer
Praise be to Allah, and blessings and peace be upon the Messenger of Allah:
It is permissible for a man to pray in a vest or undershirt if it is clean and covers the `Awrah. Men only have to cover the area between the navel and the knee when praying. This is with regard to the validity of the prayer .
But with regard to what is better and more perfect, it is better for the Muslim to adorn himself for prayer and to wear nice clothes, because Allah says (interpretation of the meaning):
“O Children of Adam! Take your adornment (by wearing your clean clothes) while praying.” [Al-A`raf 7:31]
Also, the Prophet (blessings and peace of Allah be upon him) said: “When one of you prays, let him wear his two garments , for Allah is more deserving that you should adorn yourself for Him.” (Narrated by Al-Bayhaqi, 2/235; and classed as authentic by Al-Albani in As-Silsilah As-Sahihah, 1369)
And Allah knows best.
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