The cliff hanger I left off with in monday's session of Trail of Cthulhu was the realization that a monster was creeping up behind her, and she's got a long hard sprint back to the boat. On the plus side, the PC did spend Cthulhu Mythos to figure out what the monster was, which also clued her in to a possible weapon against it if she can just escape this encounter. It cost her 4 Stability and 1 Sanity to pull that clue out of Cthulhu's butt, and she hasn't even seen the monster yet. Which is creeping up behind her.
Prior to this session, I was a little dubious about the Investigative Ability called Cthulhu Mythos. It seemed to me like a cludge, a way for the Keeper to skip out on making a well-layered mystery with interwoven clues. I understood that some GMs might need that (he says, looking down his nose as if he himself never had a plot hole, deus ex machina, or mystery that was beyond the PCs ability to solve), but it struck me as cheap. Now I understand the real purpose of it, and I wish the rules had spelled it out just a tiny bit more clearly.
As the GM, you never want to set up your mystery so that PCs have to spend Cthulhu Mythos to put clues together - that'd be cruel and unfair since doing so saps their Stability pretty hard. Neither is it intended to let PCs skip doing legwork and jump ahead of the plot. Instead the Cthulhu Mythos spend exists solely as an escape valve for PCs, who have already jumped ahead of the plot by dumb luck or clever intuition.
Normal police procedures involve raids and arrests, but that's not always an option in Cthulhu. Cops just have to figure out whodunnit, then call in the cavalry. A Cthulhu investigator has to figure out whodunnit, whatdunnit, whether or not their weapons can hurt it, how to banish it without going crazy in the process. You're on their own, and you have to work fast - or else the world might end! Once you know whodunnit, there's a lot of motivation to confront them before they can grow any stronger (or summon something stronger). When you've figured out who the villain is earlier than the GM anticipated you would, you may no longer feel any need to track down seemingly unrelated clues. In Cthulhu, though, those unrelated clues are what usually hides the means to foil the the cultists plans, defeat the monster, etc.
If, in your rush to solve the mystery you've put yourself in dangerous proximity to some nasty thing without being properly prepared for it, Cthulhu Mythos is the emergency shut-off valve. Either it gives you the weapon you need, or suggests where you can find that weapon, or points back to the line of clues you skipped over that will eventually lead to that weapon, or it just hits your sanity enough that you faint and don't have to worry about the monster for the moment.
In other words, you don't spend Cthulhu Mythos because you aren't getting anywhere - you spend Cthullu Mythos because you got too far, too fast.