I played this game any times and every time we have to advance the active mystery by one no one knows how to handle a case like this:

Ancient one: Azathoth

Active mystery: Omen of devastation

Text: When an investigator closes a gate of current omen, he may spend 1 Clue to place 1 Eldritch token on this card. At the end of mythos, if Eldritch token=players, solve this mystery.

OK so we run into "advance the active Mystery by 1"

Should we place:

an eldritch token using

If the active Mystery requires one or more tokens to be placed on the card, place one token of that type on the card.
• Clues, Gates, and Monsters placed on the active Mystery in this way are drawn from the Clue pool, Gate stack, and Monster cup, respectively.

or

a clue using

If the active Mystery requires an investigator to spend one or more Clues, place one Clue from the Clue pool on the card. Any investigator may spend Clues placed on the active Mystery when resolving an effect of that card.

This problem is common to ~25 mythos cards out of ~60 in total

I played this game any times and every time we have to advance the active mystery by one no one knows how to handle a case like this:

Ancient one: Azathoth

Active mystery: Omen of devastation

Text: When an investigator closes a gate of current omen, he may spend 1 Clue to place 1 Eldritch token on this card. At the end of mythos, if Eldritch token=players, solve this mystery.

OK so we run into "advance the active Mystery by 1"

Should we place:

an eldritch token using

If the active Mystery requires one or more tokens to be placed on the card, place one token of that type on the card.

• Clues, Gates, and Monsters placed on the active Mystery in this way are drawn from the Clue pool, Gate stack, and Monster cup, respectively.

or

a clue using

If the active Mystery requires an investigator to spend one or more Clues, place one Clue from the Clue pool on the card. Any investigator may spend Clues placed on the active Mystery when resolving an effect of that card.

This problem is common to ~25 mythos cards out of ~60 in total

In your example you would place and Eldritch token on the mystery. The rules are a bit fuzzy to some folks due to wording, but the easiest way to handle the "advance the mystery" items is to simply place one of what ever gets you closer to solving the mystery, if possible, and take the appropriate resource from game supply and not any specific player. This is to act as a free advancement of mystery. Maybe this is why it confuses some folks, as this game rarely if ever "helps" one win ;-)

Hope this helps.

​EDITED FOR GRAMMAR AND PUNCTUATION

When would this even trigger?

If the active Mystery requires an investigator to spend one or more Clues, place one Clue from the Clue pool on the card. Any investigator may spend Clues placed on the active Mystery when resolving an effect of that card.

since all mythos cards that even have a wording "spend clue" end up in placing a eldritch or gate on the mystery.

When would this even trigger?

If the active Mystery requires an investigator to spend one or more Clues, place one Clue from the Clue pool on the card. Any investigator may spend Clues placed on the active Mystery when resolving an effect of that card.

since all mythos cards that even have a wording "spend clue" end up in placing a eldritch or gate on the mystery.

Just off the top of my head the "Voice of Azathoth" Mystery requires you to spend (# of Investigator) Clues and discard the Massa di Requiem per Shuggay Artifact to solve it. There is no incremental progress to track (or advance) with that mystery. There are probably others like that.

Here's the official answer. It uses the "Hunting the Thousand" mystery as an example, that one has monster tokens vs clues.
If you're interested, here's some reasoning behind why does it work like this.

> Just off the top of my head the "Voice of Azathoth" Mystery requires you to spend
> (# of Investigator) Clues and discard the Massa di Requiem per Shuggay Artifact to
> solve it. There is no incremental progress to track (or advance) with that mystery.
> There are probably others like that.

I also found "Crown of the serpent" would be in the same spirit.

However just because that rule applies to these two, does not mean it won't also apply to "Omen of devastation" in the OP, since you are required to pay clues when resolving an effect of that card.

There are around 25 mysteries using "spend a clue to put-a-token/gate/fight-a-monster-and-place-token" mechanic. So on average one in three.

I have no problem in placing a token when it's a clean token collecting like in "Fortifying the barrier" (Syzygy).

My problem is that both of them apply and the problem just like in the Matrix is choice what applies more.

When I was writing all of this, tsumas534 answer popped up - so the lead has a choice to handle it one way or the other. OK.

For "Crown of the Serpent" and "Voice of Azathoth" you advance by placing a clue on the mystery (the last option in the OP) - these clues are then later available to use when it comes time to finally solve the mystery. For example, in a four player game you could advance the mystery four times for "Voice of Azathoth" and then when it comes time so solve the mystery all you need is for an investigator to possess the Massa di Requiem per Shuggaya artifact - they can discard the artifact and use the four clues on the mystery.

No one was disputing that