I just realized after watching Top Gun that it's likely we haven't been engaged in a true aerial dogfight for quite some time. (I know the film is fictional btw)
There were 41 38-43 air-to-air kills by the US in the first gulf war. During the air campaign in the former Yugoslavia 9 (4 over Bosnia, 5 over Kosovo.... though one of these might actually have been by a dutch F-16 it's unclear) enemy fighters were shot down by US aircraft. As far as I know that was the last air to air kill by the US
edit: reviewed sources which lead to some different numbers popping up and downloading a bunch of DoD reports so I probably made a list somewhere:
Gulf War Air Power Survey
Gulf War Toolbox
Air Force Historical Research Agency, Aerial Victory Credits site
further edit: there was an air war over libya also but I don't know the numbers for that and I just realized you asked how many were fired. I couldn't find any statistics on hit percentage..
In addition to the Gulf War and Yugoslavian theater, there were the two Gulf of Sidra incidents, in '81 and '89.
In '81 two USN F-14's shot down two Libyan Su-22's and in '89 two Libyan Mig-23's were splashed, also by USN Tomcats.
Also, if you're specifically looking for missile kills, you might want to delve deeper into the Gulf 1 numbers. I don't think US fighters made a gun-kill in that war, but I do know of an F-111 that manouever-killed an Iraqi fighter. It's likely included in the tally, but it wasn't by a missile.