Current AS&SH House Rules

Attribute Generation
Roll 3d6 in order to generate your first array. Your second array is the inverse of this (18 and 3, 17 and 4, 16 and 5, etc.). Choose whichever array you want. Make sure you qualify for the class you want to play.

This house rule is taken from Esoteric Enterprises (aff link), and it's supposed to provide a nice middle ground between 3d6 in order, 4d6 drop lowest, and 3d6 arrange to taste. I personally haven't tested it; I just assume Cavegirl did.

Hit Points per Level
Upon attaining a new level, either roll a single HD (plus CON modifier) and add the resulting amount to your total HP, or reroll all your HD (and CON modifier times your new level) to determine your new HP total. If the reroll would result in no increase in HP, increase your total HP by 1 instead.

This is a house rule I have been using for years now. It has worked fine so far. The intent is to alleviate the impact of really bad hit point rolls in the long run.

Scroll Use
Spellcasters can identify scrolls of any tradition. They can also use scrolls of any spell level. When they use a scroll of a different tradition, there is a 2-in-6 chance of miscasting (1 random target, 2 half efficacy, 3 half duration, 4 lose 1d4 random ability points, 5 opposite effect, 6 different spell of same level).

This one's a bit trickier. I haven't actually tested it at all. Since AS&SH uses so many different spell schools, the scrolls in most OSR modules would need to be tweaked, otherwise only Magicians and Clerics would have a chance to find usable scrolls (let alone learn new spells). This house rule is basically my lazy way of handling that instead of rerolling every scroll found on the fly using the tables in the rulebook, while also helping out casters in situations where I actually do roll on those tables.

Encumbrance
Encumbrance limits are calculated based on one’s Constitution score.

This one we haven't tested yet, although The Nighmares Underneath (aff link) already does this. It makes more sense in the fiction (because encumbrance isn't really about raw strength but endurance, plus STR already has so much going on for it.

Critical Hits
If an attack roll is a natural 20, use the standard critical hit table in the rulebook. Furthermore, critical hits add a notch on the target’s armour (or shield, if they so choose).

Critical Misses
If an attack roll is a natural 1, the attacker chooses among the following options:
1) Weapon gets a notch (-1 to-hit and damage per notch; breaks at 3 notches)
2) Attacker is disarmed (weapons falls 2d6 feet away in a random direction)
3) Attacker gets off-balance (-2 AC penalty for 1 round)
Unarmed attacks always result in off-balance, while ranged attacks always result in a notch.

Weapon notches have been part of my game for a really long time (about 30 sessions, probably more), and they have proven to be a very nice addition. The intention is threefold: (1) to incentivise bringing more weapons along, while balancing encumbrance; (2) to surprise the GM and the players at the table at times; (3) and to provide another resource to juggle, one which is decoupled from levelling. Repairing weapons costs one-third of its gold value per notch, which is especially expensive for magical weapons (although they are able to withstand more notches in general). High quality weapons are easy to model in mechanical terms without introduction to-hit and damage bonuses, too. It also generates anecdotes, like the fight with a bunch of skeletons, where the party suffered like 10 notches and had to retreat immediately, even though they had plenty of HP and spells. Even trivial fights may deplete precious resources now, which is a huge positive for me. As for the other options, I have tried similar outcomes in the past, but this is the first iteration I find actually fitting. This way every attack has a possible negative outcome, but there is also player choice involved for most cases.