  In this paper a new classification of monomial curves in
$\mathbb{A}^4(\mathbbmss{k})$ is given. Our classification relies on the
detection of those binomials and monomials that have to appear in every system
of binomial generators of the defining ideal of the monomial curve; these
special binomials and monomials are called indispensable in the literature.
This way to proceed has the advantage of producing a natural necessary and
sufficient condition for the definining ideal of a monomial curve in
$\mathbb{A}^4(\mathbbmss{k})$ to have a unique minimal system of binomial
generators. Furthermore, some other interesting results on more general classes
of binomial ideals with unique minimal system of binomial generators are
obtained.
