definately depends on the individual horse. 3 out of 4 of my horses right now are standardred. one is retired and no way she would be suitable for anything other than light trail riding, very long back, slender build, high head carriage naturally. she is just a pet. one is about to retire (at age 14) and i hope to find him a home as a pet or light riding only, again he is a very slender built and long back (with a slight roach hes always had), i cant see him being comfortable with working hard under saddle. both are pacing bred and always trot in turnout.
my third standardbred is still racing (only 5) and i cant wait until he retires so i can ride him, short body, very compact, very sturdy looking horse, not the huge trot so many of than have and he canters in the paddock 
i owned 3 other standardbred geldings previous to these ones, all that i had strictly for riding, one did beginner lessons at a hunter barn (he was 16.3hh and insanely thick and warmblood look to him), one did barrel racing and the other just pleasure riding due to a previous injury.
most pacing bred horses will trot, most standardbreds can be taught to canter if they dont already.
i do know a few who do/did jumpers and even one who shows smaller hunter shows and does well.
standardbreds are one breed with many many size/conformation variations.
look at the horse as a horse as an individual not just as a standardbred 