I couldn't carry a tune in a bucket so I only record other people which means my technique varies somewhat depending on how they like to work.
However, a few things. First, even if a take seems absolutely flawless, I always ask for at least one more, preferably two, just to give some leeway when I hear things in the mix that I missed during tracking. No matter how good, I always here things.
I usually ask vocalists to try to do a full take even if they make a mistake. I'd rather punch in on a full take rather than do lots of bits and pieces. I don't always get my way and, if not, just make the best of it.
Regarding punch ins, if there's one section that is problematic, sometimes letting them do just that without worrying about getting the bits before or after right can help them hit the notes and/or timing.
As has been said, feeding vocals to a bus with the same effects applied to all lets you mix and match takes very easily.
One thing I always do is spend a lot of time on the headphone mix being heard by the vocalist. Giving them the right amounts of themselves and what they need to hear in the backing tracks can really make a difference to the quality of the performance. Many inexperienced people say "that's fine" whatever you give them...try to talk them through letting you know what helps them. This includes things like reverb on the monitor feed...it helps some people, hinders others.