The difference between music players when it comes to sound quality is somewhat subjective, and some may say it does not matter. But the fact is that the different players uses different decoders for MP3 and have different default equalizers settings, DSP plugins enabled and dithering options.
One of the most popular players, Nullsoft's Winamp, is also considered one of the best sounding players when it comes to MP3 files. The MP3 files is decoded with FhG(Fraunhaufer) and is considered one of the best when it comes to "low-bitrate" files around 128kbps.
The most mathematical correct decoder and player is said to be Apollo, but is somewhat outdated.
Foobar2000(2K) is a music player based on the mpg123 MP3 decoder.
I wanted to test it because it could use the realtime upsampling plugin from Secret Rabbit Code. My first impression was that the sound was clearer and more defined than Winamp. It was also better than the more, in my opinion, muffled ITunes player from Apple.
Secret Rabbit Code upsampler is said to be one of the best upsampler after maybe Audiomove and r8brain PRO wich cost money. For best sound quality I've used the default and said to be best upsampling algorithm "Best Sinc Interpolator" with a sample rate of 96Khz. It is also possible to go even higher with 192Khz, but I don't think it is doing much unless you got a really highend stereo system. I've notice difference between 44,1Khz and 96Khz though. It may be a Placebo effect but it sounds and feels like it is more "air" in the music, exspecially in songs with bitrate over 192kbps. It also sounds more dynamic and have the possibility to set a bit depth of 24bit or 32bit.
Foobar2000 is also a very fast and lightweight player with many advanced configuration features, and is therefore my "numero uno" music player for now.
Soon I will test MAD(Mpeg Audio Decoder) plugin for Winamp, and Neutrino for UltraPlayer.
EDIT March 2008:
I'm now using my Delta 44 soundcard as output instead of the internal soundcard, and the internal codec sample rate is set to 96Khz. It is connected to my new Cambridge 640A v.2 amp and then to my headphones. With this setup I can't hear any big difference between Foobar2000 and Winamp.