Which ferrite permeability to use ?

Doesn't matter much:

Well, at 10:1, mu_eff appears to still be going up with mu_r, even for mu_r > 300. In that case, it would still matter "much".

That is a rather long rod, for just 10uH. A tiny rod, 3mm say, 1cm long, will get you 10uH, and be capable of almost as many amperes (before saturation or melting the wire -- it will be pretty heavy wire to fill the available winding area, as shown).

If you're after a loopstick, then you do want length, and to take advantage of that length, you need higher permeability. Presumably you're also targetting lower frequencies, so MnZn ferrite, probably 500 < mu_r < 3000, is what you want. (Typically #33, 44 and 77/78, from Fair-Rite, are used.)

If you're after an RFC, then material really doesn't matter, and length is better on the short side (length/diameter of 2-4), where mu_r > 100 might as well be "infinite". If you need high Q rather than high impedance, you'll want to choose lower loss ferrite, i.e., NiZn (mu 30-125 in various formulations), which is still good at 100MHz (whereas most MnZn become dominantly lossy in the 100kHz-3MHz range).

Tim