There used not to be so much violence.

There didn't use to be so much violence.

Where is the correct position of 'not'? (How can I construct the negative interrogative?)

contrapositionWhere is the correct position of 'not'? (How can I construct the negative interrogative?)

Didn't there used to be so much violence?
or
Did there not used to be so much violence?

I'm not sure but I think it seems that 'use' is correct instead of 'used' in both of the sentences.

'Used' is theoretically not correct. However, you often see it. I would say that it's acceptable these days.

Whatever you do with negative and interrogative forms of 'used to', you are going to upset somebody

When I was at school over half a century ago, the only accepted forms in educated BrE were modal-like - Used you to ...? and I used not to ... . Such forms are not common these days, though I occasionally catch myself producing a used not to ... .

If we treat

Just to confirm fivejedjon's comment, I used both forms above without thinking about it! Perhaps I should have.

Google's Ngram Viewer shows the flip in the 1970s.

didn't use to > didn't used to

CJ