Tidal forces in turn would depend on the nature of the hole. A big one would be easy going, but OTOH getting far from one would take ages, and there'd be bigger radiation/jetsam troubles. A small hole with a steeper gradient might tear the ship to shreds and pose radiation hazards of a different sort, even when the actual pull wasn't all that severe.
But losing inertial damping would be hell in any case. Not just because of the tidal forces, but because the passengers of these ships aren't built to take the stresses of impulse accelerations, and OTOH the ships aren't built to take into account the frailty of the passengers and probably don't work well at artificially low accelerations. An escape at one-1024th impulse might stumble on overheating issues or whatnot.
As for "conventional physics", our VOY heroes appeared surprised that their sensors weren't easily going through the event horizon of that singularity in "Parallax". If normal singularities don't pose problems for FTL signals, then falling beyond the event horizon of a sufficiently big and gentle black hole shouldn't be much of a problem for a starship. They could always signal for help, which would arrive in a jiffy, from their own time-dilated viewpoint, and then encase them in a warp field and propel them back to safety in no time flat.
Timo Saloniemi