What crimes can a Thane commit without punishment?

I know this is an old question, but since I have not seen a definition of what happens here it goes. Basically being a thane means that the guards will clear a bounty of up to 3000 gold once per hold. It does not matter which guard sees you or how many crimes you have committed, so long as your bounty is less than 3000 you can have it cleared by declaring thaneship when the guard attempts to arrest you. Once you have used this privilege you will need to start paying the bounty just like everybody else.

See UESP article on Bounty - Declaring Thaneship.

The crimes are:

• Murder
• Theft
• Murder of animals (e.g. chickens)
• Stealing (animals or objects)
• Pickpocketing.
• Assault (depends - sometimes you can get away with fist fighting!)
• Being a vampire (at the later stages - you get attacked on sight!)
• Transforming into a werewolf in front of people (attacked on sight)

And also, I'm pretty sure if you join the Stormcloaks, the guards of the empire will try and kill you.

There is no limit to small crimes like pickpocket, but what you might want to do if you do those small things is sheath your weapons.

PS: For lockpicking you might want to check out the perk tree because at level 40 I believe you are able to do it without being noticed.