I was wondering whether a magnet exerts any measurable gyroscopic effects. I understand that magnetism is caused by alignment of spins of electrons which have angular momentum. (I realise that that they do not actually spin though), so it seems that a magnet will have a net angular momentum direction. Is this at all perceptible?
"...a magnet will have a net angular momentum direction. Is this at all perceptible?"
As people have pointed out, it is a perceptible effect that has been measured.
Furthermore, very important techniques exploit this intrinsic relationship between angular momentum and magnetism. One of them is nuclear magnetic resonance, which uses the precession (that arises due to the angular momenta of nuclei) of spins under a magnetic field to obtain information of a system. A non-trivial and useful application of this is magnetic resonance imaging.
I would like to add that this gyroscopic property of magnets might be used to develop a new generation of highly sensitive magnetometers based on macroscopic magnet precession in ferromagnetic needles (few micron sized). The reference + detailed proposal can be found here.
To give the shortes possible answer I add a second answer.
The electromagnetic induction of a moving charge in a magnetic field is based on the electron’s magnetic moment. The magnetic field turns the magnetic dipole moment of the electron in the direction of the magnetic field. The motion of the electron undergoes a - predictable and perpendicular on the two vectors of the velocity and the magnetic field - acceleration according to the cross product of this two vectors. This acceleration leads out to the emission of a photon from the electron (Bremsstrahlung).
The photon has a pulse and this pulse reduces the velocity of the electron and acts against the momentum change of the electron’s magnetic moment by the external magnetic field gyroscope effect too. This process is repeated periodically until the kinetic energy of the electron is consumed.
It is perceptible. Lorentz force is from the electrons magnetic dipole moment when electrons moving not parallel to a magnetic field. Magnetic and electric induction are based on the same phenomena too. See my paper about the internal cause of Lorentz force.... But I have to warn you, this knowledge is forgotten to a more scolastic point of view how it is teached today and the todays comments are that Lorentz force happens because it happens. See this discussion