Do angels have wings?
In Doctrine and Covenants 129:1-3 we read:
1 There are two kinds of beings in heaven, namely:Angels, who are resurrected personages, having bodies of flesh and bones—
2 For instance, Jesus said: Handle me and see, for a spirit hath not flesh and bones, as ye see me have.
3 Secondly: the spirits of just men made perfect, they who are not resurrected, but inherit the same glory.
From this we learn that angels are people, and therefore they do not have wings.
Which raises the question why they are commonly depicted as having them? This has to do with cultural differences. In our western culture we tend to be very literal. We say what we mean and we mean what we say. In other cultures, including the Jewish one, at the time of writing the accounts found in the bible they were much more poetical, much more symbolic.
When we consider the symbolic meaning of wings, we start thinking of power of movement, and messengers. Both of which fit with what angels are used for and what they can do. Thus when we read about Winged Angels we need to understand the symbolic nature of it. If you take it with a more literal mindset you generate the commonly held belief that they have wings. Because that is what the words literally say, but not how they are intended to be understood.