For an oriented $d$-manifold $M$, we can ask whether the manifold admits a Spin structure, say, if the transition functions for the tangent bundle, which take values in $SO(d)$, can be lifted to $\operatorname{Spin}(d)$ while preserving the cocycle condition on triple overlaps of coordinate charts.
For an unoriented $d$-manifold $M$, we can ask whether the manifold admits Pin$^+$ or Pin$^−$ structures (that is, lifts of transition functions to either $\operatorname{Pin}^+(d)$ or $\operatorname{Pin}^−(d)$ from $O(d)$. This is analogous to lifting the transition functions to $\operatorname{Spin}(d)$ from $SO(d)$ for the spin manifold).
If the manifold $M$ is orientable, then the conditions for Pin$^+$ or Pin$^−$ structures are the same. So it reduces to the old condition that $M$ admits a Spin structure or not.
Every $2$-dimensional manifold admits a Pin$^−$ structure, but not necessarily a Pin$^+$ structure.
Every $3$-dimensional manifold admits a Pin$^+$ structure, but not necessarily a Pin$^-$ structure.
Question: For some any other $d$, say $d = 0, 1, 4, 5$ etc. are there statements like: every $d$-dimensional manifold admits a Pin$^-$ structure? Or, every $d$-dimensional manifold admits a Pin$^+$ structure?
P.S. This question has now been asked on MathOverflow.