Originally Posted by Hida Yanra

in general, many people use cogs without freewheels in velodromes- at the speeds and inertia of normal track racing lock rings aren't as crucial as they are riding on the roads.
You mention a front brake- are you aware that no velodromes allow bikes with brakes to be ridden while track bikes are on the track?
Street/Fixie bikes are quite different than "Velodrome" bikes in every way but marketing.
If you are running a fixed gear bike without a lockring in around-town use, please don't try skid-stopping, even a little bit- it'll go badly.

So.... if you are riding a fixed gear on the street, sure- ride with a front brake, no lock ring, don't exert backward pressure on the pedals though.
if you are riding on a velodrome, probably no need for a locking (unless you are doing match sprints and such things), but your front brake will have to go.