What are the odds that my daughter's socks drawer contains exclusively single socks? Over 20 socks but not a single pair

Depends on how many pairs the drawer started with.

I'm imagining that initially there were say 100 socks in the drawer (50 distinct pairs) and each day she randomly chooses two socks (already very unlikely to be a pair) and has some chance of losing one or both.

In this scenario it does seem intuitively reasonable that when it gets down to 20 there might not be any pairs left, but I don't know how to math it. I am pretty sure that the higher the number of initial (non-overlapping) pairs, the more likely it will end (at 20) with none left, but again the math is beyond me.