Often we know that a particular request type already exists somewhere in the hierarchy of all the request types, but it can be very hard to find them. A search function would greatly ease that pain.
I second this feature request - I've created duplicate request types unnecessarily because there isn't such a search feature. This seems like a simple add and it should be done sooner than later. Here's a mock up of what the feature should look like:
What's crazy is this EXACT feature already exists in the Setup>Tickets>Tasks screen so its certainly possible to add it so that we can search for request types!
