TTs: Some time-triallists ride fixed. No reason why you couldn't ride single speed either, but you'd need to run a bigger gear (probably, for all I know you might train and knock about on an 84" gear)
Cyclocross- if you have clearance for 'cross tyres (and mud) then cyclocross is possible on singlespeed (and there are some advantages with this setup too- less stuff to get clogged with mud). You will probably need to run a smaller gear, although one of the riders from my club does it on 42x16- and does it fairly well too. Season starts around September (although there are some summer leagues too)
Hillclimbs: Yeah, but the season is short. Many clubs run their hillclimbs in Late Sept, early October. There are some events at other times.
Grass-track racing? You couldn't really do this on your knocking around bike though.
Bicycle Polo?