Does Solitude not have any available bounty quests? It would be odd that Solitude would be the only city that doesn't give out bounties. I've spoken to Falk Firebeard and the bartender at the Winking Skeever, and neither seem to give out bounties, just the quests that they are already involved in.

The quests I remember if Solitude are scripted.

I don't remember any bounty quests either.

Solitude only has one location for a bounty quest. Thus, I'll ask a few questions.

1. Are you currently on any other bounty quests?

2. Do you have any other quests pointing at Broken Oar Grotto?

3. Is Broken Oar Grotto currently clear?

If all of these are no, then try talking to Ahtar, the headsman / prison guard leader.

^ This. If you've already got 3 bounty quests in your misc tab, you won't be offered any more until you complete at least one.

I played a character as a traveling sellsword, and this became apparent after a little while.