(These examples are shamelessly pilfered from Gröbner Deformations of Hypergeometric Differential Equations by Saito, Sturmfels and Takayama, which is perhaps the place to learn computational D-module stuff.)
Holonomic: $D\cdot\{z_1\partial_2,z_2\partial_1\}$. This left ideal has characteristic ideal of dimension 2, so it is holonomic.
Non-holonomic: let $f=(z_1^3-z_2^2)$; $D\cdot\{f\partial_1+\partial f/\partial z_1, f\partial_2+\partial f/\partial z_2\}=D\cdot\{\partial_1 f,\partial_2 f\}$. This left ideal has char.ideal of dimension 3, so it isn't holonomic.
(Edit: if you want to actually compute stuff, I recommend the Dmodules package for Macaulay2; it has tools to do most D-module things, including characteristic variety/ideal, gröbner bases etc.)