  We present a space-time virtual element method for the discretization of the
heat equation, which is defined on general prismatic meshes and variable
degrees of accuracy. Strategies to handle efficiently the space-time mesh
structure are discussed. We perform convergence tests for the $h$- and
$hp$-versions of the method in case of smooth and singular solutions, and test
space-time adaptive mesh refinements driven by a residual-type error indicator.
