  We discuss multi-graded nilpotent tuples of multi-graded vector spaces which
are a generalization of graded nilpotent pairs. The multi-grading yields a
natural notion of a shape of such tuple and our main interest is to answer the
question "Is the number of multi-graded nilpotent tuples of a fixed shape, up
to base change in the homogeneous components, finite?" Our methods make use of
a translation to the class of so-called "Multi-staircase algebras" and we
classify their representation types.
