I've tried them all.
Tubular : very specialised for racing only. Lightweight rims and tyres, can be ridden flat since they are glued on. Puncture is a pain. You have to :
- find a rough location for the puncture
- unpeel the tyre round that location (or all they way round)
- unstitch the tyre
- find the hole and apply a patch
- restitch the tyre
- re-glue the tyre back onto the rim
This is a real pain, and the tyre can often end up a bit lumpy as your stitching is never as flat as it was before - hence the comment above about sending it away. Fine for the pros and top racers with support teams.
Clincher : standard puncture repair. easy to swap in a new tube if you puncture on the road. Unless you're racing on lightweight tyres, then it's worth paying a bit extra for tyres with some decent puncture protection. There's a trade-off between protection, weight and rideability.
Tubeless : pain to set up, but once properly set up are a joy to ride. Most punctures will self-seal. bigger holes can be plugged on the road and if it's really bad a tube can be used. If you do have to use a tube, then you have to remove the valve to allow the tube to be fitted, so it's a last resort, and the tyre can be very tight to fit. Again, unless you're racing seriously then tyres with a bit of puncture protection are worth fitting. I've set up tubeless myself and it's tricky, but not that difficult once you understand what you are doing. Needs rims and tyres that are tubeless compatible for road wheels.
The quality of modern clinchers can be very good and some pros are now using regular, lightweight clinchers and some are using tubeless. But regular clinchers, with a bit of puncture protection keep things nice and simple. For winter riding I like Continental GP 4 Seasons as a good compromise of light weight, puncture protection and good grip in cold, damp conditions.