There are some very small things that have been shown to spin. These are the electrons that surround the nucleus of atoms. The only case where they would not be spinning would be when considered to be at absolute zero temperature.
The electrons orbit their nucleus when in normal state matter, and at a temperature above absolute zero. When atoms are adjacent to one another, electrons may be induced to transfer between atoms. This is how we see electricity flow.
The transfer of electrons between atoms (electrical flow) induces other effects. One such effect is EMF (Electro Motive Force), or what we interpret as Magnetism.