Stand or Sit? Handheld or Stand??

Standing is better for projection because your diaphragm isn't compressed. However, you may be hearing better results not because of your position/posture but because of a shift in room acoustics.
 
Comb filtering is the biggest problem when recording vocals. It's caused by reflections off walls, floor and ceiling - generally, the closer a reflective surface is, the worse the problem. People often surround themselves with absorption to counter the effect, but often forget that the closest boundary is usually the ceiling, especially if you're standing. Hanging some rigid fiberglass from the ceiling is the solution for that.
 
As for hand-held versus stand-mounted microphones, I do like to hold the mic in my hand, as it's much more natural when trying to keep levels steady. However, it's not an option with most condenser microphones because they're subject to picking up handling noise. For a dynamic mic, though, hand-held is the way to go, IMO.

If you play a keyboard (synth not computer keyboard) then if you sit and play, then you usually sit and sing and play also. If you play guitar, then you stand and sing, and if you play drums then you have to sit and sing and play the drums. If you play bass guitar, then you probably stand and sing, unless you are drunk and then you may want to sit down and sing (if possible) before you fall down. If any other instruments, then probably the same rules apply, and if you are jumping up and down while playing a song then you better have a sturdy floor, as long as the house is rocking with you or wherever you are playing except at a funeral.

 
 

I sit (to get at the mouse and all) and use a boom stand for my vocal mic.  If I was recording a singer, they would stand.
 
Rimshot

When I worked at Capital Records in the 80's as a studio drummer, their chief engineer Charles Ferris recorded Glen Campbell's lead vocals with him in a long padded box on his back with a mic above his face with the lights out.  Almost like a coffin.  Whatever works!
 
Jimmy
 

My vocal teacher, a professional opera singer, taught me that you should stand.  Not only to keep the diaphragm free, but also to align everything above it.  Throat, neck, chest, etc...  Kinda like the curves in a brass instrument, they're meant to be in a certain position for optimal sound shape and presentation/projection.
 
Having said that, it may not matter depending on what you're singing.

You stand up and sing while crouching down a little and wiggle around to the beat. Perhaps it is called wiggle singing, I don't know for sure.