Under as a preposition can mean
1 - in a place which is directly below: he hid under the table; the coin rolled under the piano.
2 - less than: she is under thirty; he is under age; it was sold for under $100; he ran the mile in under four minutes.
Underneath as a preposition means under/beneath: she wore a long woollen cardigan underneath her jacket; he was sheltering underneath a chestnut tree.
As an adverb it also means under: he was wearing a thin shirt with nothing underneath.
Beneath as an adverb means underneath/below: he looked out of the plane at the mountains beneath.
As a preposition it means under: have you looked beneath the cooker?
He thinks it is beneath him, he thinks it is too insignificant/too unimportant for him to deal with.
Underneath is less used than under (simply a shorter version)
Basically they're all the same word, under is just a shorter version of underneath (through years of the language changing with use) Under is probably a little more flexible, so safest to use. While beneath is still a great word, it can sound out-of-place is CERTAIN sentences:
I'm beneath the bed. (odd, like you're using old words deliberately?) I'm under/underneath the bed. (ok)
“Beneath the supermarket” would sound a little strange too.