Using an obstruction based on Donaldson’s theorem, we derive strong restrictions on when a Seifert fibered space
Y
=
F
(
e
;
p
1
q
1
,
…
,
p
k
q
k
)
Y = F(e; \frac {p_1}{q_1}, \ldots , \frac {p_k}{q_k})
over an orientable base surface
F
F
can smoothly embed in
S
4
S^4
. This allows us to classify precisely when
Y
Y
smoothly embeds provided
e
>
k
/
2
e > k/2
, where
e
e
is the normalized central weight and
k
k
is the number of singular fibers. Based on these results and an analysis of the Neumann-Siebenmann invariant
μ
¯
\overline {\mu }
, we make some conjectures concerning Seifert fibered spaces which embed in
S
4
S^4
. Finally, we also provide some applications to doubly slice Montesinos links, including a classification of the smoothly doubly slice odd pretzel knots up to mutation.