If
C
C
is a smooth, complete algebraic curve of genus
g
≥
2
g\geq 2
over the complex numbers, a point
p
p
of
C
C
is subcanonical if
K
C
≅
O
C
(
(
2
g
−
2
)
p
)
K_C \cong \mathcal {O}_C\big ((2g-2)p\big )
. We study the locus
G
g
⊆
M
g
,
1
\mathcal {G}_g\subseteq \mathcal {M}_{g,1}
of pointed curves
(
C
,
p
)
(C,p)
, where
p
p
is a subcanonical point of
C
C
. Subcanonical points are Weierstrass points, and we study their associated Weierstrass gap sequences. In particular, we find the Weierstrass gap sequence at a general point of each component of
G
g
\mathcal {G}_g
and construct subcanonical points with other gap sequences as ramification points of certain cyclic covers and describe all possible gap sequences for
g
≤
6
g\leq 6
.