my last campaign ended with the players getting party wiped by a goblin camp they tried to raid. The goblins took (I think) three times as long to finish their turn as the players. After the session, I figured I should use average damage instead of damage rolls for determining outcome of the attacks of monsters...
BUT I don't feel like this is fair to the players: if a barbarian rolls a 1 on a D12 damage die, it sucks. if an enemy attacks with a D12, they automatically deal 6 damage (+ modifiers). I could also see it the other way around, if a player rolls higher than the average.

Is there any set standard for when to use, and when not to use average damage for an enemy's attack?

I'd usually always use rolled damage unless there's a lot of attacks happening and all the rolling would slow down the game too much. So really only average in large groups.

In general monsters using average damage favors the players -- the average damage is rounded down, and randomness favors the weaker side of a conflict, which is almost always the monsters. There are edge cases where this isn't true (for example, for goblins, rolling damage is marginally better for PCs with 6-8 hp, marginally worse for PCs with 9-10 hp), but it rarely makes a big difference, luck in attack rolls is a lot more significant than luck in damage rolls.

I always roll damage. 

I come from the old days when there was no (printed) optional rule to use average damage.

Agreeing with the above.

The chaos of battle is part of the fun. When a Goblin rolls a Nat 20, but rolls 1's for damage, that is the kind of emotional roller coaster that keeps everyone glued to their seats. Without that kind of risk/reward ebb and flow, I'd personally get a little bored.

If Goblins TPK a player party, then either

• (A) The encounter wasn't properly balanced. ( 6 goblins in a group is more dangerous than 6 individual goblins.)
• (B) The players weren't prepared. (Forgot to take a rest or weren't using their resources correctly.)
• (C) The players refused to flee. (Losing is an option, sometimes running is the smart thing to do.)
• (D) It was just bad luck. ("It's better to be lucky than good.")

If you want your games to lean toward realism, then let the dice roll.
If you want your games to be "story-forward", then plan some contingencies, so that death is less likely.

Intelligent monsters might have reasons for keeping a party of adventurers alive:

• keeping them fresh to eat later,
• waiting to see if anyone will pay ransom, 
• they're searching for someone in particular, so they keep the party unconscious until their employer can inspect them,
• etc...

Unintelligent monsters are more difficult... maybe give the players a thematically appropriate "get out of jail free" card that lets them avoid a TPK once. As long as one player survives, they have a means of resurrecting the others, which can either be played out, or turned into a cut-scene. Both of which can turn into an adventure hook. Resurrection costs a lot of gold, so someone is footing the bill and expects a return on their investment...

I use fixed damage when I play in-person, to make combat resolution quicker. On a VTT, the roll is to-hit and damage at once.

Quote from account256 >>

I use fixed damage when I play in-person, to make combat resolution quicker. On a VTT, the roll is to-hit and damage at once.

 You can roll to hit and damage at once when playing live, too. 🙂

Another vote for rolling for damage every time. And if time is an issue, as I said above, roll to hit and damage together. I usually don’t as a player, because I’m a bit superstitious and don’t want to count my chickens. But as a DM, it can help move things along. 

I have colored metal d20's, so I've gotten into the habit of rolling my attacks all at once and resolving them in a consistant order according to Red, Yellow, Green, Blue. That lets attacks be resolved in just 2 rolls.

Basically the official rule is: whatever you prefer.

Many people enjoy the randomness of rolling. " After these dice go clickety-clack, now take twenty damage from the attack!" - However, if you have a situation where you've got a lot of enemies and/or NPCs all involved in the encounter, using the averages rather than rolling can quickly speed things up - especially when things like upcasted fireballs and breath attacks and cone of colds and so on are all being used. The use of averages there, printed, is a DM tool for expedience if necessary/preferred.

I most often roll the damage. If for some reason there are a lot of rolls needed, then I use the average.

But it really scares the crap out of the players when you roll all the damage dice from 6 enemies at once.

I use the fixed damage when the monsters are making a whack ton of attacks that don’t generally do a lot of damage. When doing that I periodically do average + 1 to 2 just to mix it up a bit for the players. For a lower level fight against a bunch of Gobbos, I would roll, but if the party is 5th level and up, I tend to love more to the average just to speed things up. After all, to actually challenge a party of that level, I need waayyy too many goblins to bother rolling all of that extra rolling for what will likely aggregate to approximately average anyway.

I always roll for “boss monsters” since they’re supposed to be special. I also always use rolled damage for fights with fewer monsters since they usually have fewer, stronger attacks. Those fights have fewer rolls anyway, so it doesn’t really take much extra time, and the individual attacks will stand out to the players more than the aggregate at that point.

Quote from IamSposta >>

After all, to actually challenge a party of that level, I need waayyy too many goblins to bother rolling all of that extra rolling for what will likely aggregate to approximately average anyway.

Eh, roll 10d20 and count hits, get 3 hits, roll 3d6+6.

Quote from Memnosyne >>

Agreeing with the above.

The chaos of battle is part of the fun. When a Goblin rolls a Nat 20, but rolls 1's for damage, that is the kind of emotional roller coaster that keeps everyone glued to their seats. Without that kind of risk/reward ebb and flow, I'd personally get a little bored.

If Goblins TPK a player party, then either

• (A) The encounter wasn't properly balanced. ( 6 goblins in a group is more dangerous than 6 individual goblins.)
• (B) The players weren't prepared. (Forgot to take a rest or weren't using their resources correctly.)
• (C) The players refused to flee. (Losing is an option, sometimes running is the smart thing to do.)
• (D) It was just bad luck. ("It's better to be lucky than good.")

If you want your games to lean toward realism, then let the dice roll.
If you want your games to be "story-forward", then plan some contingencies, so that death is less likely.

Intelligent monsters might have reasons for keeping a party of adventurers alive:

• keeping them fresh to eat later,
• waiting to see if anyone will pay ransom, 
• they're searching for someone in particular, so they keep the party unconscious until their employer can inspect them,
• etc...

Unintelligent monsters are more difficult... maybe give the players a thematically appropriate "get out of jail free" card that lets them avoid a TPK once. As long as one player survives, they have a means of resurrecting the others, which can either be played out, or turned into a cut-scene. Both of which can turn into an adventure hook. Resurrection costs a lot of gold, so someone is footing the bill and expects a return on their investment...

 This

Depends on the situation but in a scenario where we have a limited amount of time, like at the game shop before it closes, I'll just use fixed damage so we can get through stuff quicker. 

If there's no time limit, I roll.

Quote from Pantagruel666 >>

Quote from IamSposta >>

After all, to actually challenge a party of that level, I need waayyy too many goblins to bother rolling all of that extra rolling for what will likely aggregate to approximately average anyway.

Eh, roll 10d20 and count hits, get 3 hits, roll 3d6+6.

 That works I suppose, but when there are 7 PCs, and 10 Goblins per PC, eff that noise, the hits just do their 6 damage each. (I round damage up against PCs and down against monsters just to up the difficulty another notch.)

I use average damage for:

• Player character levels 1 and 2, to stop combat being too swingy.

• Player-controlled creatures (for exmaple, familiars, summons), to speed up play at the table.

• Any time I've got lots of creatures in play (for example, a horde of mephits dive bombing the party),again to speed up play at the table.

I use rolled damage for everything else, because I like rolling lots of dice. :-)

Anything the Players control they roll at my table. They like to roll doc too and I’ma not be the one to take that away from them on something like that.

Quote from IamSposta >>

Anything the Players control they roll at my table. They like to roll doc too and I’ma not be the one to take that away from them on something like that.

I feel the same, so it makes me sad when I tell players they are not allowed to roll.

On the other hand, when the shepherd druid has 8 wolves and a familiar, and the wizard has 8 mephits…

Just to be clear, I always let my players roll. Just when there's a time crunch I use average for the monsters.