Zombies: I'm going to argue here that a typical arming sword or any other medieval type sword is going to be much better here than a rapier. Poking holes in a dead person just isn't a convincing tactic, and if they just keep coming, a rapier's reach isn't going to help much.
Ogre: The rapier is arguably not a bad choice here. A torso kill is certainly possible. Still, I would prefer a typical cut-and-thrust sword in case I need to target their limbs or try to deflect or parry. Really, I want a zwiehander or some kind of spear in this situation. This is also a good situation for the heavy crossbow or even a human-portable arbalest of large size.
Giant scorpion: I don't want to use a rapier to fence with a giant scorpion. I don't even want to think about it. Spears, tridents, long axes, crossbows, longbows all seem like incredibly better choices. If I do need to use a blade, a typical arming sword would be okay but I would really prefer a scimitar or falchion with some reach, because I need something I can fend with. I really want to lop off the stinger entirely before I get close.
Golem: A rapier is not going to help you here. Some kind of two-handed maul seems like the only thing that is really going to get some purchase here. Warhammer/war pick would also be okay, if I had some way to get close. If I had a longsword and nothing else, I might turn it around and try to use the murderstroke (using the pommel as a mace, basically) to try to get some momentum. Bow is probably not going to do much here, but a heavy crossbow or arbalest might be able to damage some golem types.
Frost giant: LOL. Any type of melee here seems crazy, but the rapier seems especially crazy. There is no way to lunge and stab a giant through the heart, so I hate the rapier for this. Definitely I want something with a lot of hacking or smashing potential. Realistically, a shield doesn't seem like it would help much, but maybe a magic shield would be somewhat effective.