  The classical Gelfand-Tsetlin formulas provide a basis in terms of tableaux
and an explicit action of the generators of $\mathfrak{gl} (n)$ for every
irreducible finite-dimensional $\mathfrak{gl} (n)$-module. These formulas can
be used to define a $\mathfrak{gl} (n)$-module structure on some
infinite-dimensional modules - the so-called generic Gelfand-Tsetlin modules.
The generic Gelfand-Tsetlin modules are convenient to work with since for every
generic tableau there exists a unique irreducible generic Gelfand-Tsetlin
module containing this tableau as a basis element. In this paper we initiate
the systematic study of a large class of non-generic Gelfand-Tsetlin modules -
the class of $1$-singular Gelfand-Tsetlin modules. An explicit tableaux
realization and the action of $\mathfrak{gl} (n)$ on these modules is provided
using a new construction which we call derivative tableaux. Our construction of
$1$-singular modules provides a large family of new irreducible Gelfand-Tsetlin
modules of $\mathfrak{gl} (n)$, and is a part of the classification of all such
irreducible modules for $n=3$.
