Legendary Resistance can be used against any saving throw, at the DM's discretion.
RAW, there is nothing to prevent the creature, such as an adult red dragon (MM, pg. 98), from using legendary resistance against a scrying spell. Everything we need to know about legendary resistance is contained in its description:
Legendary Resistance (3/day). If the dragon fails a saving throw, it can choose to succeed instead.
The only condition for using it is stated: "if the dragon fails a saving throw". Since scrying triggers a saving throw, the dragon first can make the saving throw; on a failure, the dragon can use legendary resistance.
This answer gives a good perspective on legendary resistance that may help realign your perspective of the feature:
Legendary Resistance is a DM tool, not an ability that a monster chooses to use.
I think it's best to treat Legendary Resistance as a game-mechanical way for the DM to make a climactic battle entertaining. A monster with Legendary Resistance doesn't choose whether to use it or not; the DM chooses whether having the monster make a saving throw will make the battle more entertaining or not. It's (presumably) available so that epic monsters that are typically fought solo aren't quickly overcome by a couple of effects that require saving throws.
Ultimately, it is up to the DM to use it or not, but the DM would be entirely within RAW to use it against scrying. Personally, the party scrying on a big bad is a treasure trove of plot development material, so I would be inclined to allow it to push the plot forward.