Are water waves (i.e. on the surface of the ocean) longitudinal or transverse?

Each point is moving according to:
$x(t) = x_0 + a e^{-y_0/l} \cos(k x_0+\omega t)$
$y(t) = y_0 + a e^{-y_0/l} \sin(k x_0+\omega t)$

With $x_0,y_0$ -- "motion centre" for each particle, $a$ -- the amplitude, $l$ -- decay length with depth.

So you have exact "circular" superposition of longitudinal and transverse waves.

In deep waters, the fluid particles describe circles when a wave passes by. So, in a sense, these waves are neither transverse nor longitidinal. For a demonstration, see for example Howard Georgi's book (chapter 11).

In very shallow waters the particles go essentially back and forth. In the intermediate cases they follow eliptical trajectories.

The ocean waves are usually called "surface" waves. Whatever a particle trajectory is, the deeper in water, the smaller its amplitude. Several lengths below surface the water is still.

However deep inside there may be volume waves - from submarines, for example. They are detectable.