In EDIT you'll find PlaneCut move the cutting plane to the desired position using its widget. Clicking on the wide transparent blue arrow toggles the direction of what will be discarded.
If run in a situation were a global cut doesn't work (e.g. cutting one ear of the bunny) there's PlaneCut based on a selection too. Roughly select the area to be cut

and run SELECT/Edit/PlaneCut.
Fix you want to split the object along that plane set Slice as CutType.

Make sure that the cutting plane hits selected faces all around- otherwise MM can't fill the cut. Now hitting Accept you will not see some changes as both parts are still in one object. Run EDIT/SeparateShells to get two objects.
For a non planar cut you need to use EDIT/GenerateComplex. This tool is based on face groups so we need to define such a group as a first step.
Go to SELECT and select the area to be split off.

Now run Modify/SmoothBoundary. This smooths the selection boundary and creates a face group (if checked).

Accept and go to EDIT/GenerateComplex.
Hovering over the boundary of the group this boundary gets orange. Double click it.
You should see a new surface now.

Accept. (Note: A Complex object is a MM native feature so you'll run in troubles if you export it without splitting the Complex) A Complex seam is indicated with a green line

Run EDIT/SplittComplex to get two solid objects from it. You might hide one of the results via the eye toggle in the object browser.
