TylerE, on 2019-March-20, 11:25, said:
3♠ does NOT, repeat NOT, promise any extras. It's a mandatory shape showing bid.
I'd continue with a cue of 4♦, which doesn't promise extras either, although it would hint that partners splinter wasn't bad news. (Unlike a hand were you held something like ♠ KQJ where you have a lot of points tied up in a suit that isn't very helpful).
This is all completely correct. I also want to add, that in the modern game, cuebids show first or second round controls if below game. So 4d is either an ace, king, singleton, or void. More importantly, you are sending subtle indirect messages; 1. you can't cuebid clubs, so you don't have first or second round control in clubs and 2. you are interested in moving toward slam.
All of this should make north excited to move forward. After 4d, I'd bid 4s to show the splinter is a void. This would also promise a club control, as with no club control you wouldn't bid past game.