What kind of shoes they used to wear? - где ошибка?

What kind of shoes did they use to wear?

This sentence is Special question. We form Special question with the help of interrogative words. We must use auxiliary verbs in interrogative sentence. The construction of the Special question is —

Interrogative words + auxiliary verb + subject + verb.

 The sentence is in Past Indefinite. The auxiliary verb of Past Indefinite is to do, which takes its second form — did.