Singlespeed cyclocross is a beautiful thing, as very little can go wrong. A good chainline makes chain suck irrelevant and you never never ever stack because you missed a shift going into a barrier and forgot to get off your bike. With a 135 spaced rear hub built without dish, the probability of tacoing your wheel is extremely low.

Being a competitive singlespeeder, to me, means finishing at as well as (or at least nearly as well as, and sometimes much better than) you would have on a geared bike. This obviously can be very course-dependent (see below). If you want to be competitive as a singlespeeder in cross, may I suggest the following:

Be a good or great runner. There will be parts of the race that you can ride in the beginning that you will be forced to run toward the end. To be competitive, you need to be able to run as fast as or faster than those who choose to ride that part of the course. You should also be able to run faster than those around you on the run-ups. On anything that points uphill, you need to be going faster than anyone around you, because you will get your butt whipped on any downhills that are smooth or long enough to allow pedaling. If you are one of those people who walk up the run-ups the last few laps, you will get eaten up and spit out on a singlespeed.

Choose a good gear. This obviously is the key. You want to chose a gear that lets you not get dropped too badly on the flats and allows you to ride as much as possible on the hills. If you have to get off your bike you had better be able to run well (see above). I find that a 2:1 is best for me, although I could see going a bit higher as I get faster. Travis Brown runs a much higher gear than 2:1, but he is a machine and a half. Being a mere machine, 34:17 works well for me, although I am going to try some races in the 34:16 soon. I think that, if you are running anything less than 2:1, you need to be able to spin like a mad bastard, consistently through the whole race.

The keys to most singlespeed cross races are the start, and the end. (and the middle too). Most large-field cross races (UCI style) start on a big flat open area like a track or a paved road so that there are no bottlenecks by the time you get down to the skinnier trails. The start is important. Spin like a freak, and then spin faster. If you don't, you will be in a poor position, and you will probably get caught up in the inevitable bottlenecks anyway and loose a ton of time.

The end is key as well, because this is when everyone around you starts to zone out and spin a tiny gear on everything. You don't have this luxury. The lack of gears makes you a better cross racer at the end of the race. Make sure you are fit enough to maintain a good effort through the end of the race, or you lose your key advantage over those on gears.