Cricket. Yes noisy!

I thought maybe it was some type of cricket but it's so tiny and it sort of trills instead of chirping like a cricket.

Crickets start out very small. After all, the eggs that they hatch from are teensy.

Do the immature ones chirp? I'm too lazy to google. :)

There could be bigger ones making the noise, but it's not a bad sound...

That's a male, so I'd guess that he is the source of your serenades.

You have good eyes!!:) seriously how do you know it's male?
He made a buzzy sound. At first I thought it was an electrical sound but everytime I went into the kitchen it stopped so I then thought aha it's an insect. Sure enough little buzzer was under the sink mat. Is it a ground cricket? I've read that they are tiny and they trill instead of chirping.

Male because of the prominent cerci (needle-like things coming out of the rear end) and because it makes noise using wing features that females lack.

It is possible that being under the mat altered the type of sound that cricket made by limiting the wing movements.