I can’t get arrested in Whiterun

In Whiterun, I had a small bounty (5). Once I was bestowed a title by the Jarl for killing a dragon, I was able to flaunt my title to a guard to get him to waive my bounty.

You can get a perk in speech that allows you to bribe guards so you don't have your goods taken from you.

You can also go to the guard barracks of whatever town you're in and in the basement is a chest with your stolen stuff in it.

You can also buy a house and stash it there or give all your stolen stuff to a follower then go to jail.

When you go to jail, everything you have on you will be taken away and stored in a chest until you serve out your prison sentence.

However, you will always manage to sneak a single lockpick with you when you enter jail so you have a chance to break out. If you choose this method, you'll have to locate the chest with your equipment to reacquire it.

Should you choose to serve your sentence, just sleep in your jail cell. Keep in mind this has some consequences

The reason why the amulet was not taken is due to the fact that it was most likely not considered stolen. When you kill someone and take the stuff from their body, its not considered thievery.

Note, this is only for normal arrests/jail sentences. There is at least 1 quest that gets you sent to jail and that is treated differently.