We study regularity of general and axisymmetric weak solutions of
the 3D MHD equations with dissipation and resistance. A general weak
solution is shown to be smooth if it satisfies a Serrin condition.
The regularity of axisymmetric weak solutions is analyzed through
the MHD equations in cylindrical coordinates, whose concrete form
is derived here using Gibbs' notion of dyadic product. We establish
that it is sufficient to impose conditions on certain components
(in cylindrical coordinates) of an axisymmetric weak solution
in order for the solution to be regular.