Relativistic space rocks. Are they possible? I was thinking that the Universe is full of extreme events. Colliding galaxies, exploding stars, colliding planets (like in one of models of our Moon formation), colliding black holes... 
Can it be possible, that such event would accelerate some rocks to let's say $0.1c$ ?
Do we observe such objects? If not, why?
 A: The most likely way for a rock to reach that speed a long distance from any stars would be as a result of a close encounter with a close binary system - or even better, with a pair of black holes in close orbit.    Think of the rock as a baseball and one of the pair of black holes as a baseball bat.  The rock could fall in from lightyears away, swoop around the "bat",  and fly away with a net speed increase of twice the "bat's" orbital velocity.
A: Yes, if a rock is on a sling shot trajectory around a black hole, or a neutron star. However that speed would be relative to that black hole, or the neutron star. The speed of that rock can be very different relative to earth. I think spaghettification would not be an issue at .1c but actual calculations need to be done. And even if it is an issue at .1c, there can be rocks that can survive it.
This is not likely to happen as a result of explosive events like supernova etc. Because that causes a sudden acceleration and will likely vaporize the rock or crush it into dust. 
A: Suppose the total energy release of a typical supernova is on the order of 1044 J, for a baseline.  The question then becomes, if 100% of that energy goes into hurtling a physical object, how massive would the object be if the ejection speed was 0.1c?
At 100% efficiency (which is entirely unrealistic), the mass of the ejecta reaching 0.1c would be ~2 x 1029 kg or ~0.11 solar masses (or ~116 Jovian masses).  This is also unrealistically small for a remnant core of a supernova.
Let's suppose we had a hypernova at 1046 J, just for comparison.  Again, even with the unrealistic assumption of 100% efficiency the mass of the ejecta reaching 0.1c would be ~2 x 1031 kg or ~11 solar masses.  This is more reasonable in that most cores of supernova are larger than one solar mass but the issue is with the assumption of 100% efficiency.  Most of the energy in a supernova goes into the neutrinos and only ~1% goes into the blast waves that would accelerate the ejecta.  So at 1% efficiency we are back at the original estimate for a supernova.
Suppose we invert this and impose a net energy imparted to one solar mass and find the maximum speed of the ejecta.  For 100% efficiency and 1046 J, the speed of the ejecta could reach ~0.32c.  If we are more realistic and assume 1% efficiency, then the net speed would be ~0.033c or ~10,000 km/s, which is still extremely fast for a one solar mass object.  For a regular supernova at 1% efficiency, the ejecta speed is even smaller at ~1000 km/s, which is much closer to observed ejecta speeds as noted in other answers and comments.
Update
I wrote a slightly more detailed comment to the following question: Can supernovas or other explosive events release relativistic bombs?
