Better to use coil relay or remanent core relay for hi-fi audio switching?
There is capacitance between the coil and the contacts, and current variation in the coil will also create an AC magnetic field around the relay. So you should make sure the voltage you use to feed the coils is well regulated and without ripple. No need to overdo it, a LM317 is fine, or you can use the +15V for the opamps to feed a +12V relay with some RC filtering to make sure the switching spikes don't get into the opamp supply. But don't use unregulated DC from the smoothing caps.
Likewise, make sure noise from the microcontroller doesn't couple through the IOs and into the relays. You can do that easily with a RC filter, or even safer, put the micro to sleep with the clock off and a wake-on-pin-change interrupt to wake it up when required. If your micro is powered by a 5V switching converter, it might not be a good idea to use that 5V for the relay coils, as the switching noise could couple into the signal.
You can use signal/reed relays, make sure it doesn't have a wetting current spec, as the contacts that have such a spec require some current for cleaning.
I'd use bistable relays just to use less idle power, but that depends on cost. Reeds also use very little power.
You can use FET switches too. Their RdsON varies with voltage, and that can cause distortion. If they're on the + input of an opamp that draws negligible current, then changes in RdsON won't cause distortion. If they're on the - input of an inverting opamp, there will be no change in voltage, so again, no problem. If you use an opto-FET then it'll have constant Vgs so no RdsON variation (but that's more expensive than CMOS switches).
They do have some capacitance though, which can cause crosstalk, so you should check if that's a problem, and maybe use some SPDT FET switches to short the unused inputs, or something like that.