Subsurface oceans in satellites are pretty common: Europa, Enceladus, Ganymede, Callisto, maybe Pluto... This is due to tidal heating of their host planet, Jupiter and Saturn, which heats up the inner ice of those satellites. However, planets don't exhibit this inner ice layer, so they don't usually have subsurface oceans (except Pluto or Ceres, if you can call them "planets"). Why is that? Only small bodies like satellites present this inner layer of ice? Is there any Earth-type exoplanets that exhibit this inner layer of ice that could potentially melt down to liquid water? And if there is, why some cold planets have inner ice layers and others don't?
Some hypothesize that the Earth did have a subsurface ocean during the Cryogenian period, which lasted from 720 to 635 million years ago. The Cryogenian saw the two greatest known ice ages in the Earth's history, the Sturtian and Marinoan glaciations. There is some evidence that the Earth was completely covered with ice and snow during those glaciations. (There is also some evidence that it was not.)
Whether even older periods in the Earth's history also succumbed to snowball Earth episodes is even more debatable. The evidence has been wiped out by a billion plus years of tectonic activity. That even older periods of the Earth's history than the Cryogenian did suffer snowball Earth episodes does however make sense.
The Sun is considerably more luminous now than it was shortly after the Earth first formed. Once the Earth cooled from its formation (and that appears to have happened fairly quickly, in a geological sense of "quickly"), that faint young Sun should have resulted in a cool Earth. That the young Earth had periods where it obviously wasn't covered with ice and snow from pole to pole is the faint young Sun paradox. The apparent paradox is almost uniformly explained away via greenhouse gases. But which ones?
That said, once plants started converting carbon dioxide into oxygen, and once the oxygen stopped combining with iron to form most of the world's iron ore deposits, the greenhouse effect that kept the young Earth from freezing over should have dropped significantly. There are some signs that this happened, some that it didn't. Puzzling out what happened well before the Cambrian has always been problematic because rocks that old are hard to find.
The terrestrial planets are Mercury, Venus, Earth and Mars. Mercury and Venus are too hot for liquid water to exist at any level, Mars has lost nearly all its water and Earth has a surface ocean, not a subsurface one. The inner planets lost most of their volatiles (including water) as they formed, the water on Earth was provided by later icy asteroid impacts.
So none of the terrestrial planets have a sub-surface ocean. The other planets are gas and ice giants. Uranus and Neptune likely have liquid layers surrounding their cores, composed of water, ammonia and other "ices"
To get a subsurface ocean you need a planet that is beyond the frost line (the distance from the sun at which ice is stable in space) and in our solar system the planets beyond the frost line are either dwarfs or giants.
In a sense, the Earth does have a subsurface ocean, only it isn't a water ocean, it is an ocean of molten iron. The outer core of the Earth is highly fluid; it's no more viscous than water.
Among exoplanets, there are several candidate ice planets. Wikipedia lists OGLE-2005-BLG-390Lb, OGLE-2013-BLG-0341L b and MOA-2007-BLG-192Lb. (The principle way of discovering small planets that orbit far from their host star is by microlensing events, hence many of the candidate planets were found by the Optical Gravitational Lens Experiment, or OGLE)
There are no terrestrial planets with subsurface oceans because of differentiation. Denser materials move toward the center of the body. Iron is denser than rock which is denser than water which is denser than ice. The icy surface of these moons and dwarf planets is essentially floating on water which is floating on rock. You can actually see this on Earth. We have a partial subsurface ocean at the poles where water ice floats on top of the ocean.
As far as I know satellite data from mars observers show significant amounts of ice blow the south pole and pointers to a similar though smaller amount at the northern pole of mars. This is not surprising - Mars' plate tectonics stopped approx 1.2 to 1.5 billion years ago after the planets core cooled down enough. Mars' smaller mass and volume didn't provide enough "insulation" to keep the core hot. Thus the magnetic field of mars pretty much vanished - and was the only protection for the atmosphere against the solar wind. So Mars' atmosphere is now only as thin as earths atmosphere in 48 km height - and easily allows evaporation of surface water, so the only remaining water is sub surface OR in some shadowed craters