  In this paper, we use the Ap\'ery table of the numerical semigroup associated
to an affine monomial curve in order to characterize arithmetic properties and
invariants of its tangent cone. In particular, we precise the shape of the
Ap\'ery table of a numerical semigroup of embedding dimension 3, when the
tangent cone of its monomial curve is Buchsbaum or 2-Buchsbaum, and give new
proofs for two conjectures raised by V. Sapko (Commun. Algebra {29}:4759-4773,
2001) and Y. H. Shen (Commun. Algebra {39}:1922-1940, 2001). We also provide a
new simple proof in the case of monomial curves for Sally's conjecture (Numbers
of Generators of Ideals in Local Rings, 1978) that the Hilbert function of a
one-dimensional Cohen-Macaulay ring with embedding dimension three is
non-decreasing. Finally, we obtain that monomial curves of embedding dimension
4 whose tangent cones are Buchsbaum, and also monomial curves of any embedding
dimensions whose numerical semigroups are balanced, have non-decreasing Hilbert
functions. Numerous examples are provided to illustrate the results, most of
them computed by using the NumericalSgps package of GAP (Delgado et al.,
NumericalSgps-a GAP package, 2006).
