I think the idea on grid stoppers is that their function is to dampen any parasitic oscillations that may arise because of the tiny capacitance between the grid and the other tube elements. Usually, these are more of a problem at radio frequencies, but it is good practice to provide against them at AF, as tiny little RF oscillations can tend to add a harshness to the sound. Also, tiny little RF oscillations have a tendency to become amplified in an amplifier, and create problems well beyond the audible ones!
Having said all that, then, the idea is to use a non-inductive resistor to do this job; good practice says to keep the lead of this part short where it is attached to the grid, for even a straight piece of wire has some inductance, and you are trying to avoid inductance; a tiny bit of inductance added to the tiny bit of capacitance may actually make a better oscillator, and you're trying to stop that, after all!
The most common non-inductive resistor is the plain, old carbon-composition resistor. Film-type resistors, either carbon or metal, tend to be constructed by etching away the film in order to arrive at the desired value. The etching commonly leaves a tiny spiral of film as the final resistive element. IOW, a coil, which is going to have some inductance at some frequency, in theory anyway.
So, good practice for grid-stoppers dictates that you want a non-inductive part attached to the grid to dampen any oscillation that may arise. A carbon-comp is the most commonly available choice for this duty; certainly there are others. Another choice, if you happen to have a fast scope, would be to leave the grid-stopper out entirely, and use the scope to see if you even have any parasitic oscillations! Of course, you would have to check for this every time you changed tubes, and twice on Sundays, since the inter-electrode capacitance may change over time or due to operating conditions!
No doubt, Volt Second is absolutely correct in his assessment of resistors for this duty; it is highly likely that the type of part is just not that critical, especially since there may not be an oscillation to stop in the first place! But, a facet of good design is to provide against even the unlikely occurrence, and good practice says a carbon-comp is the best choice for this duty. Bottlehead is pretty consistent where good design and good practice are concerned!
In the past, I've used grid-stoppers up to 10 kOhms in pre-amp circuits; the amount of current running through them would generally be measured in micro-amps, if you could measure any, so the value is not all that critical, either. Plate-stoppers are another matter, of course! Both are "in the signal path", since the signal runs right through them, but there is such a tiny amount of current at the grid that any effect of this resistor on the signal has to be correspondingly tiny. Again, the plate-stopper, since it has the plate-current running through it, should theoretically have a much more profound effect.
Of course, in this hobby, the only way to really find out what you like in a given application is to try everything!
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