How to win at Shithead
The short version: win the swap before the game starts, spend low early, hold your magic for the endgame, and remember that the face-up cards you bank at the start are the hand you'll be holding when it matters. Shithead is luck wearing a strategy costume, but the costume fits better on some players than others.
The swap is the game
Your three face-up cards come back into your hand at the endgame, after the deck is dead, when you can't draw your way out of trouble. So bank your best: tens, twos, aces, an 8 if you're feeling theatrical. Beginners keep their good cards in hand and enjoy a lovely mid-game followed by a funeral.
Early game: be boring
While the deck lasts, you refill to three every turn, so there is no such thing as wasting a card. Shed your threes, fours and fives onto anything you legally can. Every low card you dump now is one that can't strand you later.
Mid game: count what's gone
Watch the burns. A pile that's been torched twice has taken tens, and possibly a four of a kind, out of the game for good. Watch which court cards have been and gone, and watch who picked up the monster pile: that player is now rich in rubbish, and rubbish comes back out eventually.
The 9 is a knife, not a doorstop
Don't spend a 9 on a small pile just because you legally can. A 9 played onto a tall pile, at a player who has been strutting around on kings and aces, is the single most profitable card in the game. Nines answered with nines create a chain of descending misery; try to be the player who starts it, not the one it reaches.
Endgame: respect the blinds
The blind cards are pure luck, so your job is to reach them with an empty hand and a burned or tiny pile. If you're first onto your blinds while others still hold pickups, the odds are lovely. And when a blind card fails, take your medicine cheerfully: the pile you inherit is at least made of cards you've seen.
Playing the table
The must-play rule means nobody can hide, so the real reads are simpler: who hoards, who panics, who always keeps a 2 for the exit. Feed the hoarder height, feed the panicker nines, and never, ever tell the table how many tens are left unless it helps you. And when the blind cards betray you anyway, remember the Harrogate maxim: it's only luck when you lose.