Just like any regular speaker, you want a subwoofer that can output linear (flat) frequency response. If you want to be able to hear absolutely everything in a recording, your subwoofer needs to be able to play down to 20Hz. You also want it to play with as little distortion as possible. And finally, you want it to be able to play loud enough to pressurize your room to the desired SPL while not giving up linearity, extension or low distortion.
That last bit is where things get really tough. There are quite a few subwoofers these days that can output linear, low distortion bass with extension down to 20Hz (or at least, close to it). But many, many subwoofers can only do this at moderate volume levels. At higher loudness, most subs cannot maintain their linearity and instead of the frequency response looking like a flat line, it starts to look like a slope.
This isn't the end of the world. For one thing, the room interacts with bass A LOT. In any room, you get what is called, "room gain", where the bass that is basically bouncing around the walls, ceiling and floor will increase the perceived loudness. And most people do not listen to their systems at full "reference" volume levels, which calls for 105-115dB "peaks" from time to time. Most of the time, most people keep the volume level quite a bit lower.
Regardless, the goals that I mentioned still stand. In the sort of price range that you are talking about, I have not come across any subwoofer that performs and measures better than the SVSound PB12-NSD. It can play remarkably loud while maintaining relatively low distortion and astonishingly linear frequency response. It is a little bit over your $500 mark, but I truly cannot recommend any other subwoofer more highly. It would be well worth your while to stretch your budget just a little bit
Be sure to really take in the physical size of this subwoofer. Among enthusiasts, the PB12-NSD is considered to be relatively small. But to most "normal" people, it is freakin' huge! lol
As I mentioned, the bass that you hear does not just come from the subwoofer alone. Your room plays a huge role in the final bass that you hear. "Room gain" might help in terms of pure loudness, but the big downside is that all of that reflected sound is actually "distortion" in the strictest sense. It is an echo and it can serve to "muddy" the sound and make it less clear and distinct.
Some of the reflected sound waves will also "double up" as they cross past one another or they will cancel each other out as the "crest" of one sound-wave crosses paths with the "valley" of another. These areas where the reflected sound waves interact strongly are called "bass peaks" and "bass nulls" and they are entirely dependant on your room. What you want to do is position your seat and your subwoofer very carefully so that - at your seat - you are not sitting directly in a "peak" or a "null". The "trick" to figuring out where best to put your subwoofer is to do the following:
- move your seat out of the way and put the subwoofer where you would normally sit.
- play some loud, constant bass.
This Test Tone CD from RealTraps is an excellent source!
- you probably have a few places in your room where you would be willing to put the subwoofer. Go to those locations and kneel so that your head is at the same height it would be when you're seated in your regular seat.
- Listen at each location and try to find the best spot where the bass is smoothest and most even - without any big peaks or dips at certain frequencies. You won't be able to escape them all, but try to find the best spot overall.
- Now put your subwoofer in that spot and move your seat back to its normal position and enjoy your better bass!
Finally, a VITAL thing to do is to decouple your subwoofer from the floor. Sound does not just travel through the air. Sound also travels through the solid structure of your home. Whether it be wood, drywall, concrete, steal, what-have-you; sound travels through the structure of your building and bass is transmitted particularly well. This is why you (or your neighbors) can often hear the bass thumping away, even when the higher frequencies are quiet or inaudible.
At certain resonant frequencies, the structure will shake just from interacting with the sound waves in the air. Other than absorbing as many room reflections as possible using bass traps, there isn't anything that you can do about this.
But a great deal of shaking comes from the physical connection between the subwoofer and the floor when the sub is just sitting on the ground - the way most people have it. Far more shaking than you would probably imagine causes your entire structure to shake in sympathy with the subwoofer. To your neighbors or other people in the house, this results in annoying, audible bass throughout the whole building! But inside your theater, it leads to "muddy" indistinct sounding bass as well.
To greatly reduce this structure-borne sound transmission, what you need is a "shock absorber" in between the bottom of your subwoofer and the floor. We call this "decoupling". And the best decoupling device is the
Auralex GRAMMA isolation riser.
This is not a "tweak". This is a MUST HAVE. Though I heartily encourage people to listen to your system both with the GRAMMA under the sub and with the subwoofer just sitting directly on the floor. Hearing is believing in this case. And your neighbors will thank you (or rather, just not come knocking to complain

)
So we did go over your $500 mark, but the PB12-NSD atop an Auralex GRAMMA is absolutely the highest value. There is nothing that I know of at the same price or cheaper that can compare. So, in this case, spending a little over your budget is WELL worth it!
Best of luck!