What should I bid after Opener’s 2 [club] rebid? I didn’t want to leave him there with a singleton.
You are correct that having a singleton in partner’s suit is very, very bad. However, bidding is just going to get you in more trouble! Opener has shown 12-14 HCPs and at least five clubs. In this auction partner will always have at least six clubs, so the fit isn’t as bad as you think. When you don’t have a fit and don’t have the values for game, stay low and safe.
How do you know partner has at least six clubs?
Opener is unbalanced, because he did not rebid NT.
Opener does not have four spades, because he would have bid 1 [spade] with four.
Opener does not have four hearts, because he would have raised you with four.
Opener does not have four or more diamonds and only five clubs, because he would have opened 1 [diamond] with a minimum hand.
Therefore, he must have at least six clubs.