Back in February 1989 I was in my second year at Warwick University. (Pure Maths with a generous side of Computer Science, since you asked.) It was there that I first met Steve Sykes and various other friends that I am still in touch with.
For reason that are now lost in the mists of time, I compiled a set of variations on the classic “My dog’s got no nose” joke, and sent it off to rec.humor.funny, a heavily moderated newsgroup, where I was pleased to see it published.

Over the years, I accumulated several more jokes for the collection (some mine, some from friends). I had been keeping them on my old website, but now I’m moving them over here, in the hope that the fine community of Reinvigorated Commenters will be able to contribute more.
1st man: My dog’s got no nose.
2nd man: How does he smell?
1st man: Awful.
— Traditional
1st man: My dog’s got no nose.
2nd man: How does he smell?
1st man: He doesn’t. He’s got no nose.
— recovered from the mists of time by Stephen “Haldane” Sykes
1st man: My dog’s got no nose.
2nd man: How does he smell?
1st man: You poke his eyes out.
— Mike Taylor
1st man: My dog’s got no nose.
2nd man: How does he smell?
1st man: (Pause) … Well, it’s kind of hard to describe really.
— Mike Taylor
1st man: My dog’s got no nose.
2nd man: How does he smell?
1st man: (Pause) … It’s OK, he has got one really.
— Andreas Pagel
1st man: My dog’s got no nose.
2nd man: How does he smell?
1st man: (Pause) … Oh, go and get a glass of water.
— Andy Charles, incorporating material from Monty Python and the Holy Grail
1st man: My dog’s got no nose.
2nd man: How does he smell?
1st man: By means of a cunning hydraulic mechanism.
— recovered from the mists of time by Stephen “Haldane” Sykes
1st man: My dog’s got no ears.
2nd man: How does he smell?
1st man: With his nose, obviously.
— Mike Taylor
1st man: My dog’s got no nose.
2nd man: How does he smell?
1st man: Of his own accord.
— Trad., Arr. Mike Taylor
1st man: My dog’s over thirteen billion miles long.
2nd man: How does he smell?
1st man: Don’t change the subject.
— Mike Taylor
1st man: My dog’s literally doesn’t exist.
2nd man: How does he smell?
1st man: Don’t ask stupid questions.
— Mike Taylor
1st man: My wife’s gone to the West Indies.
2nd man: How does she smell?
1st man: When it’s ajar.
— Trad., Arr. Mike Taylor
1st man: How do you make a Venetian Urn?
2nd man: I don’t know, how do you make a Venetian Urn?
1st man: (Pause) … I’m sorry, I appear to have made a mistake.
— Mike Taylor
1st man: My frog’s got no nose.
2nd man: How does it smell?
1st man: Frogs can’t smell.
— Mike Taylor
1st man: My dog, who incidentally had no nose, died yesterday.
2nd man: How did he smell?
1st man: We shall never know. He died before he could tell me.
— Andy Clews
1st Restaurant owner: My dog he die yesterday.
2nd Restaurant owner: How do he smell?
1st Restaurant owner: Don’t know, but customers think he taste pretty good.
— Andy Clews; Country Of Origin Omitted Here For Sake Of International Relations
1st Circumlocutionist: I have in my possession an animal belonging to the family Canidae, and it appears that he does not possess any extra-facial olfactory organs.
2nd Circumlocutionist: Could you therefore impart to me such knowledge as may be necessary to describe how that animal circumvents the problem of satisfying his olfactory senses?
1st Circumlocutionist: Unfortunately the non-ambiguity of your enquiry does not easily permit me to provide a clever answer, but I am in fact thinking of referring the animal to an olfactologist. However, the animal does have a fairly unpleasant body odour, should you be interested.
— Andy Clews
Cannibal Dog: My dog’s got no nose.
— Mike Lessacher
Senile old fool: My dog’s got no nose.
Young delinquent: Why is that?
Senile old fool: He’s been dead for 30 years.
— Mike Lessacher
Senile old fool: My dog’s got no nose.
Young delinquent: Why is that?
Senile old fool: I’ve been dead for 30 years.
— Mike Lessacher
Cannibal 1: My brother’s got no nose.
Cannibal 2: How does he smell?
Cannibal 1: Delicious!
— Matt Wedel
Student: My dog’s got no nose.
Teacher: “My dog has no nose.”
Student: Yours too?
— Matt Wedel
Zoologist: My lungfish’s got no nose. Well, it has choanae, but they may not be homologous with those of tetrapods, and recent research indicates —
Non-zoologist: Oh shut up!
— Matt Wedel
Covert vampire: Help! I’ve got no nose!
Unsuspecting victim: Yes you do, you big ninny. Here, come look in the mirror and I’ll show you.
Covert vampire: I have to be going now.
— Matt Wedel