If I click the start menu button, and move the mouse a little bit upward so that the cursor is on the "Places" or "Applications" section, typing a key does nothing. It works if the mouse cursor stays on the start menu button or on the "Favourites" section.
I do not know whether this is an intended behaviour or a bug, but this is inconvenient. Windows' start menu does not have this problem and whenever I type something it initiates search.
I think Mate's start menu should work like that. I am on Linux 18.04 with everything up to date.