These are all the major ones. Do not attempt any of them unless you have professional experience with (a) sleeping and (b) being some kind of a cat. (h/t The Frogman, for sharing this image.)
1. The Full Situp
To achieve the full situp, you must begin with the genuine intention of exercising your abs and promptly fall asleep midway through the task. This position is extremely advanced and not recommended for amateur sleepers.
2. The Awkward Spoon
The goal here is not so much intimacy as it is the socially uncomfortable sharing of a physical space with someone. Bonus points if your arm falls asleep but you're too embarrassed to move it.
3. The Semicircle
Tuck your tail between your legs and imagine that you are an omelet.
4. The Sunbather
The trick is to look like someone who is acting comfortable whilst also appearing extremely uncomfortable. Let's take this excellent opportunity to coin the term “meta-comfortable.”
5. The Double Bed
You will need a partner for this one. The goal is not so much comfort as an expression of sheer, unadulterated greed.
6. The Half-Box
Any old box will do, but two of your feet – preferably on opposite sides of your body – must remain outside the container at all times.
7. The Backstroker
Do not even attempt unless you have tiny, tiny, precious little legs.
8. The Sleeping Baby
Find a baby. Imitate the baby.
9. The Fur Pile
For this, you will need at least three friends who are not averse to your sleeping on them.
10. The Full-Box
Just get your whole damn body in there no matter what it takes. Be the box.
11. The Drunken Radiator
Just because you are obviously some kind of gin-addled hobo doesn't mean you can't be nice and warm.
12. The Sleeping Dog
Find a dog. Imitate the dog.
13. The Librarian
Bury your furry little head in your paws and try to look as contemplative and bookish as possible before drifting off.
14. The Ruler
Measure the floor with every inch of your tiny body.
15. The Windowsill
The whole world is your hammock.
16. The Clothes Dryer
Imagine that you are a wet T-shirt, fresh from the washing machine. Drape yourself accordingly.
17. The Pot Luck
Think of yourself as a last-minute fruit salad that everyone will be very polite about but probably not enjoy all that much.
18. The Head-Rush
Head to the ground, paws in the air – let gravity do the rest.
19. The Odd One Out
For this one you will need first to find two willing conformists.
20. The Mid-Sentence
Only recommended for individuals with extreme forms of narcolepsy.
21. The Bag Of Limbs (Box Edition)
Have a friend or loved one take you apart and put you back together haphazardly inside a box.
22. The Bag Of Limbs (Couch Edition)
Same as above, except (obviously) without the box.
23. The Dog Bed
Not a bed for dogs, but a bed that is made of dogs. I.e., the most comfortable bed you will ever sleep on that also smells kind of funky.
24. The Office Worker
Fall asleep on the job. LOL.
25. The Married Couple
Don't be afraid to snore.