Critical fails (natural ones on attacks)...are they a thing? I'll usually say it's a huge whiff, but I like to keep the game serious-ish and not turn expert fighters into bumblers. I'm interested to see how most other people handle them.

I do a mix of flavor and mechanical setbacks; whatever fits the story and situation.

Recently, I had a Halfling player fire her hand crossbow at a night hag.  She rolled a 1 and that made her crossbow string snap.

I also had an Elf player try to climb up the side of some very narrow stairs like the Halfling player, she rolled a 1 and I had her slip on her wizard's robes and tumble down the stairs taking 1 point of bludgeoning damage.

It always depends. Sometimes I have them hit themselves, but then a whole battalion of archers accidentally shot themselves to death, and a swarm of crawling claws instantly tore themselves apart.

Sometimes they hit an ally or a weapon breaks, but only something minor.

Other times they just miss, or get disadvantage on their next attack roll.

Honestly I've never liked the idea of critical failure actually having a mechanal set back because you basicly punish people for making more attacks, because by this sort of system it means a level 20 fighter who is the peak of material power and can potentially make 8 attacks in a single round is more likely to critically mess up then a level 1 fighter who is at the starting point of their potential and is barely stronger then your average guard 

But that's just me and the minor flavor of "big miss" I think works perfectly fine 

@jimmop Agreed. It also favors spellcasters who use saving throws for their attacks (e.g. Sacred Flame) and can't "crit fail."

Generally, there’s little to differentiate a 1 from most rolls under 10 in most situations unless the modifier is substantial. I those cases it’s generally a whiff or I say the player is distracted by something else to focus on their tactics. With a 1, I ALWAYS spice it up because it is a Critical Fail no matter what the modifier is. It’s horrendous to see, so I make the most of it and elaborate the hysterical goings ons of the event. DnD is all about fun. A failure is enough punishment without creating setbacks. Unless the players want a real challenge, I don’t punish them with those types of obstacles or equipment wear. 

My pistol in the Eberron setting has a Misfire chance, which means on a 1, the prototype weapon my Warforged artificer created jams, and if I attempt to fix it during combat and fail, it breaks entirely. To me, it's fair on two accounts. One, the range on it is pretty good, and it does decent damage, so it balances out that risk/reward scenario, and two, it's thematically appropriate, as the early gunpowder weapons were not known for their reliability.
Outside of that particular weapon, most crit fails in my groups are handled with a little joke about the pathetic nature of the attempt, but without punishment.

Varies.

Depends on the difficulty of the attempt and desired result based on environmental factors.

At worst:
A difficult attempt in a dangerous environment has a serious consequence - a fumble as an art form. Nobody in the vicinity is immune (except by immunity stats).
At best:
An easy task in a calm environment could just be a whiff (or disrupting the calm at the most).

Small mechanical setbacks are fine, or the DM causing something to happen. I personally haven't had a good time when having to roll on a critical failure table.

I usually do something silly for critical fails. Just flavor to make a joke, though nothing mechanical. A few times I might make things mechanical. Like an archer getting a nat 1 on an attack roll, their arrow splinters and is unrecoverable after the battle. Stuff like that, nothing too bad. 

It's an extremely situational subject.  Trying to make a blanket statement ruling on it will inevitably fail, because there are so many variables.

How my DM handled it is that when you Crit Fail, you roll the to-hit die again.  If it comes up under 10, something happens.  If it's a ranged attack, it might hit someone else in the line of fire (which might even be an ok thing if it hits another enemy), or in melee if there's another target in range, friend or foe.  If there is no other target, then nothing happens.  None of this "you swing, your hand slips, and you punch yourself in the face" nonsense.

Basically, you have to fail twice for a Crit Fail to occur, and even if you do it's only a bad thing if you had already failed at group positioning in the first place.  So effectively 3 times.

Just for myself, I tend to only reference the 'critical fail' bit in a role playing sense, not a mechanical one. Most often I describe some sort aspect of failure if the character is trying to do something somewhat outrageous.

a 5% chance of a serious setback is just too much. Any setback should at most be similar in power to a critical hit, preferably less. Otherwise it just feels bad. In a live game i play the DM uses really punishing nat 1  rules so it just made all of the casters use DC based spells because when you have  5% chance of killing one of your party members it's just better to not cast attack spells.

Also, is this involving the often-used-but-technically-house rule of additional crits/fumbles on Checks and Saves? (It's just more fun that way in my opinion.)

...or is this just for Attack Rolls (and a few other specific situations like Death Rolls) as specified in RAW?

I've played in games that have used all the answers in this poll, and generally I prefer to just treat it as a bad miss.

If the players do want to run with fumble rules, I prefer the Pathfinder method of making them roll again. If they beat the DC of what they were trying to do or the AC of who they were trying to hit, it's just a silly miss. If they fail the check again, then it's a randomly determined mechanical set back.

Again, I only use these rules if all the PCs want to use them.

it really depends on the context but for me its usually just flavor. However, should my rogue get caught pickpocketing a guard or should my bard fail to seduce the mimic things may get ugly.

Should my bard fail to seduce the mimic...

So, when your character tries to seduce a monster, you mean?