I just wrote a REST API for my company and used PUT and DELETE (Tomcat doesn't support PATCH yet).
Plenty of DELETE in Stripe's API: https://stripe.com/docs/api#delete_recipient
Github uses HEAD, PATCH, PUT, and DELETE: http://developer.github.com/v3/#http-verbs
Twilio supports PUT and DELETE: http://www.twilio.com/docs/api/rest/request
There are all darlings of the HK community with highly praised, widely used REST APIs. Have you read through the developer docs for most APIs?
(edit: typo)