While doing an image search, I found these images of propellers:
and
Some appear to be straight-ish on the leading edge and curved on the trailing, some curved on the leading edge and straght-ish on the trailing, and symmetrically curved and some pretty squared off.
It seems like, well, a propeller is a propeller. They all need to push the air back in a way as to propel (hence the name?) the plane forward.
Wouldn't there be a mathmatically 'best' design?
The best design depends on what your goal is. This changes with operating conditions:
Best power/thrust ratio
For lightly loaded propellers this would be minimum induced loss. These are the propellers on human-powered aircraft, windmills or motor gliders. Minimum induced loss is achieved by distributing circulation elliptically across the propeller disc, and a typical consequence is a small chord near the tips and a low number of propeller blades. If you want to research this more, I recommend to play around with XRotor by Mark Drela (and maybe reading this). Mark used it to design the MLE (below, NASA photo EC87-0014-8) and Dedalus propeller.
Maximum thrust at high speed
Highly loaded propellers, which can be found on large turboprop aircraft, are less concerned with induced loss, but need to put out as much thrust as possible with a limited prop area. The diameter is restricted to limit tip speeds, and the tips typically are mildly supersonic. To limit the suction peaks on the blades, the local lift coefficient needs to be low, which is achieved by increasing chord and the number of blades. Increasingly, now tip sweep is also used, but this drives up the torsional loads on the blades and leads to warpage in operation. An extreme example is the P-27 prop on the D-27 turboshaft of the An-70 transport (below, picture by Marina Lystseva).
Ship propellers
They operate in a medium which is 800 times denser than air, and the main concern is to avoid cavitation. This again means to limit suction peaks and leads to very high blade chords. To limit draft, ship propellers need to have extremely small diameters.
Submarine propellers
For submarines, add minimum noise emission to the design goals. Minimum draft is not a concern, so the propeller blades are less stubby. Their sweep distributes the cutting through the wake of the rudders over time, which helps a lot to reduce noise.
The fancy shapes of some model aircraft propellers have more to do with marketing or unscientific voodoo beliefs by some hobbyists. Generally, a straight shape works best.
A couple of extra points for piston engine aircraft:
The size and amount of propeller blades mostly depends on the power output of the engine. A giant prop on a tiny engine won't be good, because it consumes too much energy just to spin it. Conversely, a small prop on a big engine would waste a lot of the engine output, as the prop would overspeed before the engine reached peak performance.
Matching prop size to engine is one of the challenges for designers. As mentioned in the previous answer, it depends on what the aircraft will be used for.
Also, most propellers do not actually move the aircraft by 'pushing' the air behind. If that were the case it would not make any sense to have them on the leading edge of the wing or on the nose - you should place them with clear air behind. The propeller is actually an aerofoil, producing lift using the same principle as a wing. Negative pressure is created on the 'upward' side of the prop, so the propeller 'pulls' the aircraft forward.
