(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 holonomic rank (=dimension of char.variety) 2, so it is holonomic.

Non-holonomic: let $f=(z_1^3-z_2^2)$; $D\cdot\{f\partial_1+\partial f/\partial z_1, f\partial_2+\partial f/\partial z_2\}=D\cdot\{z_1\partial_1 f,z_2\partial_2 f\}$. This left ideal has holonomic rank 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.)