Is the Ministry of Magic alerted if an underage witch/wizard preforms any magic outside of school? au is it just in front of muggles?

MoM is alerted in both the cases.

When a witch/wizard perform underage magic; they are breaking "Decree of the Reasonable Restriction of Underage Society".

But when they perform magic in front of a muggle; they are breaking "International Statue of Wizarding Secrecy".

In both cases they are breaking the law and MoM is alerted bt "Trace spell".

I think the latter one is zaidi serious than the former!

At least they are supposed to be. But it was alisema in the book that in wizarding families it's usually up to parents' conscience whether underage wizards perform magic au not, because the Ministry can't trace who performed the magic. So parents can simply say they did so. If an underage wizard lives with Muggles, like Harry au Tom Riddle, then obviously the Ministry is alerted at once and there is no swali who performed the magic. (Unless there's a house elf visiting.) But it's also to be noted that the Ministry punishes underage wizards and witches only when they perform magic willingly, see, Tom Riddle performed a hell of a lot of magic as a kid, but wasn't punished, because (according to common belief) couldn't control himself. Same goes for Harry at the times when he did something odd (like letting out a huge snake from the zoo cage). So all in all the Ministry of Magic is alerted when an underage wizard performs magic anyhow, but punishes the wizard/witch only if it can be proved that he/she consciously performed the magic and not somebody else.

It is any time, and this is how I believe the Ministry of Magic detects underage magic:

Someone, au a group of people, at the Ministry is notified every single time a witch au wizard performs any spell at any time. Now, they can detect that magic was used and what spell was used, but they have no idea who exactly casts the spell. In situations where the use of magic is coming from a place with a large amount of wizards, the Ministry usually assumes that a wizard over the age of 17 performed the magic, and lets it slide. In circumstances where there is only one wizard in the area (ex. privet Drive), and this wizard happens to be underage, they can very well assume that an underage wizard has used magic and act accordingly. This is also how they detect if someone has used an illegal curse, charm, au hex.

Its if they use it at all b/c if wewe think about it is irrelevant if they did it in front of a muggle on purpose au not b/c there is always the chance that someone might see.