It's quite common.
I had Garlic Prawns (shrimps) in a faily exclusive restaurant in Darling Harbour (Sydney) last weekend, with shells still attached to the tail.
From what I can gather it's mostly about aesthetics.
I suspect there is some truth in the idea that it gives the impression of more shrimp for your money as well.
Some also argue that it adds flavour during roasting or sauteeing.
It should also be noted that many people will in fact eat the tail.
(The entire shrimp is edible.)
It's not my preference, so a simple pinch at the base of the shell and the shrimp will slide right out.
(They usually - or at least should - provide you with a small finger bowl filled with water, and sometimes a slice of lemon, when you're expected to remove the tail shells yourself.)
cooking.stackexchange.com/users/533/Stuart Helwig
You do get some flavor benefits from cooking with the tail on, just like you do by cooking meat on the bone vs. removing the bones.
It makes the shrimp more shrimpy which helps the shrimp stand up to rubust flavors like those found in Fra Diavole.
As for when its appropriate to serve tails on, I've always used size of the shrimp and final application be the judge.
If the final application is to be mixed in to something else, like rice or pasta, I remove the the tails, but if the shrimp are to be laid on top of something, I would typically leave the tails on.
It is alway approiate to leave tails on if the shrimp are large and not a good idea if the shrimp are small, as tail removal is hard to do in small shrimp after cooking.
Another point is that if it is intended to be consumed with the hands, you should leave the tails on as they give you a very nice handle to grasp and eat from. Soups and anything else that is intended to be eaten with out a knife beside it should be served tail off to facilitate easy consumption.
And it is normal for Fra Diavole to be served tails on.
cooking.stackexchange.com/users/446/sarge_smith
It's also a bit of a culture thing.
Where I was born shrimp was always served without shells or tails.
Whereas in Europe, shrimp is mostly served with tails and shells.
At first came as a bit of a shock to me, because it's more work to no perceived benefit.
But, after the initial shock, I've come to realize that very likely there is a flavor component that shell & tail add.
cooking.stackexchange.com/users/89/Vinko Vrsalovic
I might be alone here since no one else has mentioned this, but you can actually eat the shrimp tail and shell.
I generally only do if the shrimp are fried so that the tail becomes nice and crispy.
I also only do it when in the company of friends; you'll get a lot of strange looks and gasps otherwise.
cooking.stackexchange.com/users/60/hobodave
Tails, or shells?
The tail makes a great handle, but the shell/legs and all, I don't want to handle hot saucy foods with my fingers!
Since it's less expensive for the guest to remove the shells than for the chef to do it, I've learned how to remove shells from a shrimp without fingers, but I resent finishing the meal when the chef or line cook could have done it before OR after cooking it.
cooking.stackexchange.com/users/1539/JR in WV
In my experience, a popular (and similarly-named)
Mexican dish camarones a la diabla is without exception served with the tails still on.
Yes, it is kind of annoying if you don't want to eat the shrimp tails, but I think it helps make the shrimp appear more substantial (because there often aren't many of them), so they're typically left on.
cooking.stackexchange.com/users/285/nohat
I think it's popular to leave the tail on to give the impression that the shrimp is somehow fresher, larger, and more "homemade" as it was procured with the shell on and shelled on premises.
I don't believe it has anything to do with flavor, otherwise, they'd leave the head on, not just the tail.
cooking.stackexchange.com/users/948/ant
I have grown up in Fla...
we eat shrimp several times a week..
I have eaten in 5 star restaurants, and sent my order back to have the tails removed...
there is Never a good reason to leave them on in a sauce..
I do not care if you are Bobby Flay, or Morimoto or Paula Deen, or some guy doing street vendor stuff....
IT IS NEVER, NEVER, NEVER...appropriate to leave the tails on shrimp that are served whole in a sauce..
NEVER!..it might look good, but someone has to put the shrimp in their mouth, and then?...
whatt to do with the tails... pull them out of your mouth with your fingers?, poof them out on the ground?...
nice manners.. discretely spit them in to a napkin?
NO
NO
NO....
I have been a private chef for 12 years...
I have never had someone come back and tell me that my scampi would have been better with tails...
some writers here have mentioned how hard it is to get them off...
if it is?... go out for dinner...
a slight pinch at the base of the tail pops them off... and the dog loves 'em...
people dont...
Tail your shrimp going into a sauce... make your guests remember the shrimp, not what they had to do with the tails.
cooking.stackexchange.com/users/18964/Don
I ate at a local Italian dive last night and ordered  Shrimp Fra Diavolo , which included a spicy sauce served over linguini noodles.  The dish was great, but the tail was still attached to the shrimps and I was annoyed that I had to remove the tail to eat my dinner.  Is this normal?  Other than shrimp coctail, when is it appropriate to serve shrimp with the tail still attached?  Are there flavor/seasoning benefits to cooking shrimp in the tail?
When is it appropriate to serve shrimp with the tail still attached?
cooking.stackexchange.com
<shrimp><presentation><seafood>
cooking.stackexchange.com/questions/3498
cooking.stackexchange.com/users/217/Ben McCormack
This is common in some cultures and can give the impression of a larger, fresher shrimp as well as possibly improving the flavour when cooking. While shells and tails are often removed in the US, they are actually edible. It is quite common for the tail to be served on the shrimp and this is for presentation and flavour purposes. You can also eat the tail and shell.
It is quite common in Europe to leave the tail on for presentation. It also benefits the flavour. The tail and shell can be eaten.