There have been proposals to mine asteroids, perhaps after “tugging” them to a more accessible location. As well, proposals have been made to move asteroids which threaten collision with earth.
Asteroids would be much easier to deal with if they were not spinning. But the chance of a given asteroid having exactly zero angular momentum is, well, exactly zero. How to stop it spinning?
Brute force could be used with chemical rockets. But this involves transporting the mass of fuel for energy and reaction mass. Launches typically require 10 times the payload mass in fuel. That’s 10 tons of launch fuel for every ton of fuel that arrives on site at the asteroid. Yet there is already energy (sunlight) and reaction mass (rock) on site.
There is a similar problem if the reaction mass is in the form of a reaction wheel. A huge, high precision machine needs to be transported. Once angular momentum is transferred to the wheel, how is it dispersed? Using rockets gets us back to the problem of transporting fuel mass, as well as the mass of the reaction wheel.
Is there a more elegant way? Momentum shedding weights on wires? Light sails in the shape of a pinwheel? Giant radiometer vanes?
One method could be to use light sails:
A pair of spacecraft, joined by two tethers, encircles the asteroid with a snare. After carefully aligning themselves with the rotational axis, the snare is closed. The spacecraft extend masts. at the end of each mast a light sail is deployed, mirrored on one side and black on the other. The light sails are angled to slow rotation. Once rotation is stopped, the light sails are "feathered" to have no effect. Or they could be used to deliberately rotate the asteroid to sun synchronize or align to an incoming spacecraft or for temperature control.
@ontrack and @Ferrybig made a good point about using a yo-yo mechanism. These are commonly used to de-spin spacecraft which have been spin-stabilized. A yo-yo is very effective since the spin rate is decreased with the square of the “string” length. A plain-Jane yo-yo maneuver will always leave residual spin which will need to be handled with another mechanism, but this residual spin can be reduced to any arbitrary rate with a long enough string.
However, with orchestrated timing of the string feed-out, the residual spin can be used to wind the string around the spinning object, yo-yo style. Rapid retrieval of the string by the yo-yo mass could use the inertia of the yo-yo mass to halt residual rotation.
On-track’s suggestion of using mined asteroid material as the yo-yo mass gets full marks for elegance. However, that necessitates setting up a mining operation on a spinning asteroid. An alternative is to use the mass of the spacecraft's spent booster as the yo-yo mass.
The cable, winch and batteries are all located in the booster to maximize the yo-yo mass.