Its important to think about whether you are recording a choir, ensemble or a solo singer.
Other things that might be relevant is how the person is used to working. For example it might be difficult for a singer who is used to singing in a rock band to perform the same way if isolated.
If recording a group of singers or a choir, and omnidirectional mic might be a good choice with the singers making a circle round the mic.
But otherwise a cardoid mic in front of the singer could be used. It should be pointed between the nose and mouth for a balanced recording. It may be necessary to think about the room. An omnidirectional condenser mic will pick up room reverb etc. This might be a good or bad thing depending what effect you want. The distance between the singer and the mic will change the balance of room sounds versus the singer.
A dynamic microphone could also be used, e.g. the Schure SM58.
If the mic is picking up pops and other noises from the singer it might be possible to reduce this by changing the angle of the mic or by using a pop screen.
Endless possibilities. It obviously depends on whether the piano is upright or a grand/baby grand.
If a grand piano, then you can position the miss above the strings with the lid open. You can achieve different sounds depending where you put them. If you position them more towards the treble strings you get a brighter sound. Similarly if you are recording an upright you should open the lid and place microphones above the piano. They can also be pointed at the front of the piano.
Condenser or dynamic mics can be used.
Bass guitar can be recorded either through DI (direct inject) or through a miked amp or a combination of both.
If using DI method you also need to route through a DI box so there isn't an impedance mismatch. If recording an amp then you can get distortion. Valve amps are generally better for bass.
There are other challenges. Compression is usually used when recording bass - there is such a big different between the open E string and something played right up the neck on the G string. Also players may use techniques like slapping.
Choice of mic for recording bass as always depends on budget and the effect you want. If on a budget a dynamic mic close to the to an amp would be a good choice e.g. Schure SM57. But a large diaphragm condenser would be good too.
i'll blog again after Monday to capture what we actually did.