I remember a couple of years ago the people under us and above us jacked up the noise so we had it coming from both sides. It SUCKED. Now, the people above us were "running" their huge Boxer inside because they were too lazy to walk him. We could actually hear the ball bounce across the floor, followed by this massive beast chasing it down. It sounded like an army was storming a shore or something. We actually started bouncing a ball off the ceiling to mess with the dog and it confused him so much that they stopped doing that
Now, I tried not to get too upset with the people above us as they were just going about their routine and if I lived next to them, for instance, I wouldn't have heard a peep from them.
Now the people below us, that was a different story. Sketchy couple and LOUD music at ALL hours of the day and night, although they liked to start it after 11 pm a lot. Now, I know that you're going to pick up noise living in close quarters, but I knew this guy's music library inside and out- that's how loud it was. My last straw was Christmas night, so when it happened again I called the police. And you know what? No loud music anymore. All they did was tell him to turn it down (we had knocked several times on his door to ask him oursevles-kindly- but he never ever answered, so we thought maybe he would answer if it was a police officer knocking

)
Lately, there have been a couple of cases here of people being shot or hurt for calling out a neighbor, so I'm going to say to just call the cops and ask them if they wouldn't mind dropping by their apartment on their patrol. That really is the best (and most effective) way to handle it. They don't even have to know it was you. ETA: This was also the advice from our police department (Denver), so I guess they aren't bothered by such requests.
And honestly, I think they would want to comply, especially if their legal status is in question. Don't want to ruffle feathers, you know?