Option 1 - Nothing
Many fixed gear riders are short distance, and tend to be close to home. The creed is to remove superfluous things from the bike making it lighter and simpler. Why carry tools at all? All you need is a cellphone, or some way to pay for taxi.
Some tyres have a phenomenal puncture resistance, so this makes punctures less likely, at the cost of weight and/or rolling resistance. Solid tyres are the extreme version of this.
Option 2 - Sealant
If you have tubeless tyres with sealant or tubes with a puncture sealant inside, the idea is that punctures will self-seal and retain the air pressure. Personally it just slowed the leak down a bit, and wasn't worth the four-fold cost.
Option 3 - All the tools for the job
- Adjustable spanner or ring spanner that fits your bike axle nuts
- One or two tyre levers, generally plastic
- Air, either a pump (frame or mini) or a CO2 inflator and cannister.
- Spare tube(s) of the right size. These can be a patched and tested ones. I carry one for each 50 km of riding expected, so 3 for a 120 km.
- OPTIONAL Patch kit, or sticker patches - these are for when you're used or shared your tubes.
- OPTIONAL rubber or nitrile gloves, for keeping clean
- OPTIONAL wet wipes or a rag for cleanups.
- Something to carry all this in securely.
The larger pumps are likely to mount to the frame, but smaller ones can be bagged with the other tools. Under-Seat nad-bags are popular - I store mine in a high-vis grellow-coloured velcro or zip closed cloth bag, which is velcro-tied under the top tube, straight behind the headtube so its out of the way. On the tandem, its valcro-strapped to the bottom of the front seat tube, just above the keel tube, again out of the way.
You may as well add a bike minitool with all the hex drivers etc. I also carry a presta-schrader adapter, a chain masterlink and cableties in my on-bike toolkit.
Each of my bikes has the same stuff duplicated. I wouldn't want to flat on the road bike and find I brought a 26" MTB tube by mistake!
Finally, littering. Don't be that guy and leave your dead tube on the roadside. Instead take it home and patch it properly with the 10 minute glue in the comfort of home. Test it inflated overnight to 10-20 PSI naked, then empty it, roll it, and its your next on-bike spare. Tubes are fine with a dozen patches, as long as they don't leak, and a 50c patch is cheaper than a $5 tube.