How death and dying are handled in most OSR games, even between different editions of old-school D&D, varies wildly. The most popular version, B/X, has death happen immediately when a PC reaches 0 hit points (and usually even has most of your magic items destroyed, just to rub it in).
I've looked at dozens of games that want to make this slightly less lethal, more lenient, and most of them handle things the same way: You don't die immediately at 0HP. But you might just after. Roll death saves every round after dying to see if you die. Roll on a death-and-dismemberment table each time after going down to 0HP that you take damage. Die at -10 HP or some other negative number.
Immediate death at 0HP has something a lot of these solutions unfortunately throw away. There is an incredible tension you get when a single stray arrow or lucky hit has a good chance of killing you forever. The more you can reliably just drop unconscious, and suffer no lasting consequences, the less hesitant you become to solve things with your sword all the time. You'll probably be fine, right?
Now obviously that's not suited for what every system is trying to do. There are some games that I think don't need death at all - 5e manages to be simultaneously too non-lethal and too lethal; death is so rare that you can reliably get into fights to solve problems, which the game is built around and assumes you need to do, but that rarity means it never feels like a part of the intended gameplay loop. Death in 5e is a record-scratch moment of "Wait, what? That's not supposed to happen". Once you dip below a certain threshold for lethality, death becomes an interruption to the game, not an integral part of it; I think such systems would be better with no random chance of character death at all, instead using alternative consequences for losing. Because losing in 5e means dying, and dying interrupts the cool heroic-fantasy story you're trying to tell, so the GM just fudges the dice or pulls a deus ex machina out of their ass to save you and get things back to what you're actually there for.
I don't run OSR systems to be heroic fantasy. I run them to be survival horror, or at least any genre where combat is a big deal. Where risking getting stabbed is incredibly dangerous. I think it leads to more interesting games than systems where combat is something you can rely on, because murdering things is one of those hammers that makes every problem look like a nail. When combat does break out, it's tense! You could die! And so death-at-0HP seems ideal.
Unfortunately, this comes into conflict with another ideal.
It's very very common to see players shy away from OSR games with the reasoning, "How am I supposed to get invested in my character if they could die at any moment??" This is not an unfounded criticism I can handwave; I got my start with RPGs running 5e for my friends, and for all my critiques of that system, that improv-theatre part of it where we got invested in our little underdogs and their stories was the main attraction. It was the biggest reason we showed up to play, second only to getting to hang out with friends. It was fun. And addressing this is important if we want to have the same sort of fun with the OSR games I run instead of 5e now.
So how do I deal with this in my games? Where's the sweet spot between tension and character investment? Let's talk about Grave.
Grave's an odd, very nice game. It's a Knave hack designed for dark-souls games, which gives it some odd quirks that contrast typical sword-and-sorcery fare, like how all the players are undead who collect and eat physical tangible souls to level up. One of these quirks is the death system.
A PC in Grave can die at least 10 times before for-reals actually dying for good and being removed from the game. This seems absurdly generous compared to your typical OSR game, but a few points: You still die at 0HP, and critically, your extra deaths never come back. When a low-level rat scores a lucky hit and drops you in a fight, that wound never recovers. And the permanency of that means PCs still treat entering combat as a life-or-death decision. Every death accumulated is a ticking clock.
I think this is also more satisfying from a narrative/roleplay standpoint. In a traditional D&D game, including 5e, death is usually sudden and unsatisfying because it's so uncommon and out-of-nowhere. No real tension because it so rarely happens. But what if you could see your death inching closer?
Over the course of a campaign, watching a PC rack up more and more permanent scars, you can see it coming. Your death. It's a ticking clock, and both the player and their character can tell they're getting closer and closer to death's door. You could get a lot of drama out of that. Creating this sense of inevitable doom, tinged with a little hope that maybe you'll retire before the clock ticks down completely, allows a player to prepare for the loss of their character so it's not so sudden. Your final death becomes closure instead of a story cut short.
I've been thinking about how to implement this in more standard fantasy games (where you aren't playing undead that constantly revive) and I think I want to keep a little bit of a die roll, a little uncertainty, but still ensure you get a few guaranteed not-deaths for the first few times you fall in battle. So here's how I'd probably handle it:
Every time you reach 0HP, you gain 1 scar. Then, roll 1d6 + your total number of scars. If it's 10 or higher, you die. If not, you fall unconscious. Taking damage while unconscious gives you another scar and another roll as if you just dropped to 0HP again.
I love the image of a PC visibly racking up more and more scars on their body as they get close to death, especially being able to say where each one came from. Getting more world-weary and paranoid as the clock ticks down and they can feel death catching up to them. Maybe have near-death PCs get little visits from the grim reaper or something.