Which is right: dad's corner or dads' corner? I see dad's corner used regularly so I am assuming that's right, but in my mind dad's is singular whereas the corner belongs to multiple dads as it's a place for dads to go to meet.
Both are grammatically correct but they have different meanings:
A useful comparison is Father's Day (or Mother's Day). It's a day to appreciate many fathers, but is written this way as it's a usually day to appreciate one father at a time. This is a generic plural.
Another example is Dad's Army, the name of a sitcom about the Home Guard during the Second World War.