“conventional” tail: since this is the general aviation forum I assume you mean low subsonic speeds (a conventional low tail is used on supersonic planes to avoid the main wing’s shockwave). Easy to build and rather straightforward in input response. Usually it doesn’t matter if the tail plane gets some turbulence flow, which it by the way hardly ever gets since it is spaced to avoid, and if if it does it only shakes a little, won’t loose control because of that. Jetairliners don’t enter AoAs that would put the tailplane in the wings turbulences. Behaviour depends on the locaton relative to the center of gravity/it’s axes and the center of lift.
“H”-tail: Isn’t done any more except A-10 (here for reasons of survivability). Only for really low mach numbers and military applications (if you want to shoot a gun straight back w/o putting a gun in the tail). The An-225 has it to increase stability and also to be able to put a space-glider on top.
“+” tail: don’t know why these are built. on old MiGs because they wanted to be able to take off the engine and exhaust, so they put they tailplane a little higher. same problems as a “T”
“T” tail: structurally heavier since the vertical part needs to take all the load from the horizontal stabiliser, but has less drag due to the increased geometrical lenght. can get into superstall. On biz-jets because of area ruling and the engines taking up the space down below.
“V” tail: less weight, less drag, but problems in coupled roll/slip situations. Only on old Beeches and gliders.
no tail, “Nurfluegler”, flying wing: no drag, no weight, restrictions in crosswind, complex input. With manual input successfully done only by the Horten brothers in WW2 and a glider from Akaflieg Darmstadt around 1990. B-2 does it with computers.
“Delta” configuration: not in GA, only good for supersonic drag
“Canard” and trebble flying planes = decoupled surfaces: Good for drag and weight, but complex input and restricts overall airframe design. Lóok at the Piaggo or Starship.