Given a numerical semigroup
H
⊆
(
Z
≥
0
,
+
)
H\subseteq (\mathbf {Z}_{\geq 0},+)
, we consider the locus
M
g
,
1
H
\mathcal {M}_{g,1}^H
of smooth curves of genus
g
g
with a marked Weierstrass point of semigroup
H
H
. We show that for all semigroups
H
H
of genus
g
≤
6
g\leq 6
the locus
M
g
,
1
H
\mathcal {M}_{g,1}^H
is irreducible and that for all but possibly two such semigroups it is stably rational.