Let
X
=
{
X
(
t
)
,
t
∈
R
N
}
X= \{X(t), t \in \mathbb {R}^N\}
be a centered Gaussian random field with values in
R
d
\mathbb {R}^d
satisfying certain conditions and let
F
⊂
R
d
F \subset \mathbb {R}^d
be a Borel set. In our main theorem, we provide a sufficient condition for
F
F
to be polar for
X
X
, i.e.
P
(
X
(
t
)
∈
F
for some
t
∈
R
N
)
=
0
\mathbb P\big ( X(t) \in F \text { for some } t \in \mathbb {R}^N\big ) = 0
, which improves significantly the main result in Dalang et al. [Ann. Probab. 45 (2017), pp. 4700–4751], where the case of
F
F
being a singleton was considered. We provide a variety of examples of Gaussian random field for which our result is applicable. Moreover, by using our main theorem, we solve a problem on the existence of collisions of the eigenvalues of random matrices with Gaussian random field entries that was left open in Jaramillo and Nualart [Random Matrices Theory Appl. 9 (2020), p. 26] and Song et al. [J. Math. Anal. Appl. 502 (2021), p. 22].