This is a tough question, as it truly depends on the horse. Fussing with a bit could be caused from being a general grouch, not wanting to do the command asked, not trained well enough in the cue being asked, potential pain, needing their teeth floated.
Bits/bridles are also meant to be a secondary cue, not designed to be completely dependent on. So, when asking for the 'woah' it shouldn't consist completely of pulling on the reins, but also a deep heavy seat and potentially a verbal cue. It also depends on how soft/strong the hands behind the reins are.
Personally I choose to put my horses in the mildest bit they are comfortable in. Some like unbroken mouth pieces, some like single joints, some like double. I generally start with a french or lozenge double jointed bit. Assuming I'm direct reining. If they don't like double jointed I'll try mullen mouth straight mouth pieces. If they don't like that then I'll try single jointed. But typically if they don't like either of those bits they're not going to much like a single jointed. I've never found a horse who likes a single jointed more than a solid or double jointed equal bit, but I've heard they exist. As for which rings, I like full cheeks with keepers for young/green horses are the lateral aids are so clear. Lateral aids meaning, when you pull left the bit slides through their mouth a little and pushes the full cheek sides onto their opposite side of their mouth, pushing the head over.
For western I'll typically start with a Bumper Sweetwater Bit.
I find that often if I feel the need to put a stronger bit on a horse it's because I or someone who's been riding the horse has been too strong handed and the horse has learned to ignore the cues. So I go back to basic bit training, giving to the bit left and right and backing up off of very mild pressure.
If your horse is fussing a lot off of the bit a tooth floating may be in order or some retraining to giving to the bit.
Also the type of metal helps too, copper and sweet iron can help keep a horse's mouth soft, but some are very picky.