Tim Vine

So this bloke came up to me and said "describe Tim Vine here", and I said "Tim Vine is a British stand-up comedian who has gained considerable renown for his mastery of the pun. His entire act consists almost entirely of him standing up and delivering a stream of groan-worthy yet somehow amusing plays on words, with the occasional comedy song or other strange interlude". Also known for starring alongside Lee Mack in sitcom Not Going Out.

This performer provides examples of

Catch Phrase: Arguably, "so..." which is how many of his jokes start. He likely uses it because so few of his jokes have any relation to the others. If you wanted to extend it, you could argue that "so (person X) came up to me", "so I said to this bloke" (which was actually the title of one of his DVDs) and "so I went to the doctor's/airport/watch shop/wherever" are also catchphrases.
Ear Worm: "Pen behind the ear!"
George Lucas Throwback: It's not clear whether this is intentional or not, but he comes off as though he's stepped out of vaudeville.
Hurricane of Puns: Songs aside, virtually his entire act.
Hypocritical Humour: The Importance of Rehearsal.
Incredibly Lame Pun: Some of his puns are definitely lamer than others. The best ones tend to be those which have a reasonably solid internal logic.
Overly Long Gag: Pen Behind The Ear.
Pungeon Master: Perhaps the finest example of this trope in modern times.
Rapid Fire Comedy: He used to hold the Guinness World Record for Most Jokes Told In An Hour.
Self Deprecation: Some jokes:

I went into a mirror shop and I said to the bloke behind the counter "I want to buy a mirror you tall blonde idiot". He said "I'm over here sir".

Sliding Scale of Silliness Versus Seriousness: Probably the silliest it's possible for anything to ever get.
Small Reference Pools: A victim of this: many of his jokes have been wrongly attributed to Tommy Cooper.
So Unfunny Its Funny: The "funny" anecdotes in "Family Holidays".
Surreal Humour: The set-ups of some of his puns are strange enough to turn the jokes into this. For example:

I was in a restaurant and a duck came up to me and said "your eyes sparkle like the morning dew". I said "waiter, I asked for aromatic duck".