I alternated tubulars and clinchers (different wheelsets) on my commuter bike for over a year and it was clear that I had fewer punctures with tubulars.
In fact, I have so few punctures that I have to reglue tires after a year or so since the glue dries up. YMMV.
when a tubular does flat, you almost never get a catastrophic deflation (bang), you get slow leaks.
if you're careful you can ride on a flat.
I finished the last mile (uphill) of a race on a flat.
they are more comfortable, especially when cornering because of the suppleness of the side walls.
you can get a sub 1500 gram wheelset for well under $500 (Campy Record and Mavic Reflex).
Try that with clinchers.
Tubular wheelsets are generally much lighter than clinchers. if you do get a slow leak, tufo or stan's tire sealant will almost certainly fix it.
Now all of my bikes run tubulars.
For my commuter bike I use Tufos which last forever despite the broken glass that you inevitably see on roads.
For racing Veloflex are superb.
And as was mentioned tubulars dominate the cyclocross scene.
Once you get the hang of gluing, it's not really a big deal.
The main disadvantage in cyclocross is that you need multiple wheelsets to be able to shift tires on short notice due to weather changes.
Dry vs mud, say, requires two completely different treads.
Finally, for most places in the US you have little choice but to buy your tubulars off the web since the LBS is unlikely to stock them.
See: http://www.worldclasscycles.com/tubulars.htm
bicycles.stackexchange.com/users/7663/JohnS
I also got a set of tubular rims with an old vintage racing bike
and I've stuck with them.
I don't find them to be a hassle, but gluing does take a little technique and practice to be able to do it neatly.
I think most people who haven't done it are afraid of the gluing, but if you like to tinker with things its not difficult.
With sealant, I rarely have to change tires, so I find myself gluing a new rear tire about once every 2k miles.
You can find tubular road tires at bike shops and online at places like: http://www.biketiresdirect.com/search/tubular-road-tires http://www.competitivecyclist.com/components/tires.39.html and others
I'm sure.
What you'll find is there aren't as many sizes and brands as clincher tires, but there is a decent selection.
The two tires I'm riding on right now, and the two hanging in my garage have come off of eBay.
bicycles.stackexchange.com/users/4015/Ritch Melton
Tubular tires are very common in cyclecross; their construction and the way they mount to the rim make it possible to run a lower tire pressure.
bicycles.stackexchange.com/users/5161/Eric Gunnerson
Tubular tyres are still popular with racers and many others.
It is most certainly not true to say that no shop has them or that they are not used these days.
One of their most overlooked advantages is that carbon wheels for tubs are cheaper than their equivalents for clinchers.
Personally, I love riding tubulars and only use clinchers on my indoor training bicycle!
bicycles.stackexchange.com/users/7483/VaVaVeteran
I've only ridden on tubs, have not had a flat in 25 years
and I find the gluing process zen like.
Never used the tape.
I have veloflex Carbon tubs on carbon rims on my Cervelo and various tubs Vittoria, gommitalia, contis, challenge etc on 19 vintage Italian bikes that I ride.
Mind you those Italian Steels are babied, not ridden on gravel!
I am religious about tire pressure and inspecting the tubs after each ride, pulling out glass shards etc.
I also let some air out to rest the tubs after each ride.
You can check out some of my collection .
bicycles.stackexchange.com/users/25429/iRydstl
I bought an older second-hand racing bike, which has  tubular tires  mounted. I went to a bike shop and wanted to buy a spare tubular tire, but they told me:   Wow, such an old bike! No, we don't have tubular tires. No shop has them. They are not used anymore nowadays. Instead, use an inner tube an a tire.   Is that true? If so, why?
Are tubular tires still used today?
bicycles.stackexchange.com
<tire>
bicycles.stackexchange.com/questions/16893
bicycles.stackexchange.com/users/6299/Uooo
Tubular tires are still available and are popular with racers and cyclecross. Many people prefer them, one reason being they seem to puncture less often but, in the US, they usually have to be bought online as few local bike shops do not stock them.  Tubulars tend to have fewer punctures than clinchers, and are often used in cyclocross. They also have a following among racers. Disadvantages include difficulty switching tyres and finding them in local bike shops. Some people worry about the gluing, but it is not difficult with some practice. One benefit of tubulars for racers is that carbon wheels designed for them are cheaper.
Tubulars tend to have fewer punctures than clinchers, and are often used in cyclocross. They also have a following among racers. Disadvantages include difficulty switching tyres and finding them in local bike shops. Some people worry about the gluing, but it is not difficult with some practice. One benefit of tubulars for racers is that carbon wheels designed for them are cheaper.