The result from these tips will be minimal, but every bit helps, right? Especially when your neighbor sings like a cat in heat and your landlord won't let you cover the walls in foam.
Fill in Holes and Cracks
If you've got a noisy street below your apartment, check out the wall that faces that side of the building. Any air leaks are also going to be sound leaks. Check around windows and doors, but also look for gaps between your drywall and power outlets. Fix any offending leaks with caulk or weatherstripping.
Seal Off Your Door
If you're irritated by the noise in the hallway outside your apartment, the door is definitely the first place to look for a solution. Even if your corridors are indoors, install weather-stripping to help keep noise out. Also, an unobtrusive door sweep attached to the bottom of the door will help with soundproofing.
Hang a Fabric Wall
If you're hating the late-night music from a next-door neighbor, attack your shared wall. Not with a broomstick, but with fabric. Hang curtains or a gorgeous fabric on the offending wall—wall-to-wall will work best, but even a decorative fabric panel will help—to cushion the noise.
Move Your Big Furniture Around
If you're not down with the curtain-wall look, try moving your furniture layout around to get the same effect. A big, upholstered couch on your shared wall can help absorb some noise too.
Get a Big, Shaggy Rug
If the sound is coming from the apartment below, a big, plush rug might be all you need to keep the peace (and quiet). The bigger, the better; just make sure it's thick with plenty of sound-absorbing fibers. Layering rugs is also a great solution, especially if you already have wall-to-wall carpet.