The specs you need from an opamp in an audio circuit are somewhat dependent on the particular circuit. As you state, requirements of good audio circuits are high signal to noise ratio and low distortion, relative to many other analog circuits.
The noise figure of an opamp is usually a fixed value over a particular frequency range. What you care about is signal to noise ratio. You therefore need to take into account the amplitude of the signal when deciding how much additional noise is acceptable. For example, 100 µV equivalent input noise is a serious problem in a front end amplifying 1 mV microphone signals (20 dB S/N), but irrelevant on top of a 10 V power amp output driving a speaker (100 dB S/N).
Distortion is much dependent on the circuit around the opamp. The amount of negative feedback and gain headroom make a big difference. The linearity of parts, including the opamp, matter too. This is not just a spec you look up in the opamp datasheet.
The opamp bandwidth limits how much gain that amplifier stage can realize. Opamps are not very good amplifiers open loop, and depend on negative feedback to have good characteristics and be predictable. A rough rule of thumb is you want about 10x gain headroom to allow the negative feedback to do its job. Since the highest audio frequency is 20 kHz, that means the opamp needs a gain×bandwidth product of 200 kHz just to break even. 1 MHz would allow you a closed loop gain of 5, for example, and 10 MHz a gain of 50. However, note that higher bandwidth opamps have higher noise too. It's often better to get the same gain from multiple stages instead of trying to do it all in one stage.
Rail to rail I/O is not something required for audio. You should have large output swing headroom compared to the nominal output level anyway. Audio has sudden large pulses that must be reproduced without distortion. 10x more swing than the top RMS output is a good start. For example, a ±12 V output for line level audio is the minimum for serious work, although ±15 V is better.
One parameter you didn't mention is slew rate. If the opamp output can't move fast enough for those sudden pulses (like a sharp drum beat, for example), then they will be distorted. This may often be the actual limiting factor on the maximum transient output amplitude an audio circuit can handle.