Is it appropriate to call a British person a "Brit"?

It's not offensive. It's not commonly used here for ourselves, and I've never heard other Europeans use it.

According to the NOAD it's informal. But it doesn't explicitly specify about the adjective being informal.

The OALD agrees on it being informal and there is a good note on it. I'll paste the second part:

The noun Briton is used mainly in newspapers: The survivors of the avalanche included 12 Britons. It also describes the early inhabitants of Britain: the ancient Britons. Brit is informal and can sound negative. Britisher is now very old-fashioned.

It used to be used negatively in Northern Ireland ("Brits out"), and I would find it mildly offensive, but would always take the biggest cue from who said it and how it was said rather than the word itself

It's what I would use informally about myself and is becoming more common with phrases like "Brit-pop".
Briton is a bit more formal but in speech it's hard to distinguish from Britain.

It's definitely not offensive in the same way as Jap or Paki would be.

I am British, live in Britain and can state I have never heard the British use the term "Brit" about themselves. Nor have I heard any other country use it other than the Americans. We tend to say "I am British" rather than "I am a Brit". Newspapers refer to British people as "British" or "Britons" (as in "ten Britons died in the fire"). We would say "were there any British people there?" or "were there any British there?" — never "were there any Brits there?".

Offensive? Not really — more disrespectful because it suggests the Americans can't be bothered to use our terminology and have to invent their own.

I'd say Brit is about equivalent to Yank. It is usually used with slightly derogatory overtones, but it isn't very offensive (but is very informal). My (US) car registration is BRITSKI because I'm a British Skier, but 7 letters is rather limiting. I wouldn't normally use Brit; I'd not get up in arms about it, unless the rest of the context was nasty (and the 'up in arms' would be about the context more than the use of Brit).

Brit is definitely a commonly used word for describing people from Britain. While it may turn up in the occasional tabloid headline, it is mostly used by 'non-brits'.

However...

I am English and personally I don't like to be considered a 'Brit' because it has several negative connotations, one example being 'Brits Abroad' which refers to the British hooligan holidayers who travel to popular beach destinations and get drunk and disorderly.

For this reason, I prefer English. I would also suggest that citizens of the other British countries would also prefer to be referred to by their own nationality (Irish, Scottish, Welsh) due to various conflicts in the past (generally with England).

As for formality, it is definitely informal, having only come into use in the last century.

Hope this helps.

N.B. As an adjective, 'British' is more applicable since 'Brit' is the noun. The same can be done with 'Scottish' and 'Scot', with 'Scot' being a noun.

OED

Brit, n.3 and adj.

A British person; (= Briton n. 2.) Only occasionally found before the second half of the 20th cent.; in early use not a self-designation.

Note the first example is American English and not complementary. The 1904 example is Australian and, at that time, Australians were considered "British".

1884 Galveston (Texas) News 12 Sept. 3/3 Let the News make a suggestion—that the Brits call themselves Yankees.

1904 D. B. W. Sladen Playing Game i. v ‘Imperial Government! I call that too damned funny! Do you mean the Japs?’ ‘That word is most offensive to them. How would we like to be called Brits?’

The next two quotes are both from British authors:

1961 S. Price Just for Record viii. 69 Your working-class Brit is a glutton for celebrities.

2005 T. Hall Salaam Brick Lane v. 102 What does it take to break the ice with the Brits? A power drill?

Brit is informal and neutral.