Depending on the tile used, I prefer mounted. For those corals I mount, I personally use white marble tiles. This is basically pure calcium carbonate. Research and experience have shown this to be an almost ideal growing substrate. Many if not most corals will utilize elements from the substrate they are growing on, and this provides a beneficial medium. I've anecdotally seen a greatly increased rate of growth and attachment with these tiles. That's my reasoning. If it's going to be ceramic or plaster, I'd prefer unmounted, though it's easy enough to remove most frags and remount on my own tiles. Really, though, I've rarely had any problems with shipments caused by the method or materials used to mount. I've had lots of shipments with various plugs, tiles, rubble, and free, and when properly packed and healthy all arrive just fine.
More often than not, problems are the result of the conditioning of the coral frag, not the shipping. I've had corals come through all kinds of conditions and pull through. It's the corals that are freshly cut and shipped that seem to have the most problems. The double stressor of being fragged and then immediately shipped causes most of the problems, IME. I prefer to frag my corals several days to weeks ahead of time, depending on the species. This gives them some chance to heal and build energy reserves, etc, and reduces the chances of death from stress or infection.
I have no problem doing as others have suggested. I grow the frag out, frag a piece off and mount in my place of choice, and keep growing the old frag for trades and such.