Hey all, please bear with a newbie -
1. Is there a consensus on the best type of aerofoil section for fins (and number of fins) to achieve speed>mach?

less drag = more speed
drag ~ frontal surface area
fin strength ~ thickness

I'm guessing strength necessitates some optimal minimum thickness, which necessitates streamlining which violates assumption 7 associated with Barrowman's CofP calcs.
2. How does this affect CofP calcs?

Doubtful I'll ever progress to such lofty ambitions but it doesn't hurt to keep the grey cells greased.
Thanks in advance.

While a full airfoil shape is ideal for subsonic, it's a rule of diminishing returns - generally the extra effort just ain't worth it. For trans-sonic and above, it depends a lot on the interaction with pressure waves generated from the tip and other discontinuities. So no, there is no one 'ideal' form. In practice, most appear to be flat with a simple taper on both leading and trailing edges. As far as I know, the only advantage to tapering the edge on the fin tip, is to reduce the mass slightly.
3 fins is optimal, however 4 fins is still quite common.

The drag on the fins of a rocket is nothing compared to the base drag, or the pressure drag on the nose going supersonic, really the only thing you need to break the speed of sound is more power.

The number 1 thing that kills fins at high speed is flutter. It's not so easy to predict or calculate flutter but the harmonic frequency of the fin is proportional to its stiffness, the stiffer the better. If you're using composites and you design your fins for stiffness then you'll automatically meet any strength requirements you might have (unless you're using a lot of high modulus carbon in which case you might have to check ultimate strength too).

The double wedge aerofoil (pointy triangular front to pointy triangular back) is the best for supersonic flight but not for subsonic flight, so you can't really have your cake and eat it too. The planform shape of the fin isn't such a concern for aerodynamics, usually you'd choose one for looks or for practical reasons assuming you have enough fin area to keep the rocket stable. I tend to not have the trailing edge of the fin protrude past the bottom of the rocket as it increases the chances that you'll damage a fin on landing. Similarly a tapered fin has structural advantages, I would usually go for a tapered fin with no sweep. Avoid significant forward sweep because that can cause structural divergence in torsion (basically the fin tip starts twisting one way and the twist causes increasing forces, which causes it to twist more etc until it breaks).