The asymptotics is examined for solutions to the spectral problem for the Laplace operator in a
d
d
-dimensional thin, of diameter
O
(
h
)
O(h)
, spindle-shaped domain
Ω
h
\Omega ^h
with the Dirichlet condition on small, of size
h
≪
1
h\ll 1
, terminal zones
Γ
±
h
\Gamma ^h_\pm
and the Neumann condition on the remaining part of the boundary
∂
Ω
h
\partial \Omega ^h
. In the limit as
h
→
+
0
h\rightarrow +0
, an ordinary differential equation on the axis
(
−
1
,
1
)
∋
z
(-1,1)\ni z
of the spindle arises with a coefficient degenerating at the points
z
=
±
1
z=\pm 1
and moreover, without any boundary condition because the requirement on the boundedness of eigenfunctions makes the limit spectral problem well-posed. Error estimates are derived for the one-dimensional model but in the case of
d
=
3
d=3
it is necessary to construct boundary layers near the sets
Γ
±
h
\Gamma ^h_\pm
and in the case of
d
=
2
d=2
it is necessary to deal with selfadjoint extensions of the differential operator. The extension parameters depend linearly on
ln
h
\ln h
so that its eigenvalues are analytic functions in the variable
1
/
|
ln
h
|
1/|\ln h|
. As a result, in all dimensions the one-dimensional model gets the power-law accuracy
O
(
h
δ
d
)
O(h^{\delta _d})
with an exponent
δ
d
>
0
\delta _d>0
. First (the smallest) eigenvalues, positive in
Ω
h
\Omega ^h
and null in
(
−
1
,
1
)
(-1,1)
, require individual treatment. Also, infinite asymptotic series are discussed, as well as the static problem (without the spectral parameter) and related shapes of thin domains.