What is the $\mathbb{Z}_2$ cohomology of an oriented grassmannian?

I was surprised to learn that the ring structure of $H^*({\rm Gr}^+(k,n);\mathbb{Z}_2)$ seems to be unknown, in general. The ring structure in the case $k=2$ is given in

Korbaš, Július; Rusin, Tomáš, A note on the $\mathbb Z_2$-cohomology algebra of oriented Grassmann manifolds, Rend. Circ. Mat. Palermo (2) 65, No. 3, 507-517 (2016). ZBL1357.57065,

and the recent preprint https://arxiv.org/abs/1712.00284 by Basu and Chakraborty gives partial results in the case $k=3$.

The main tool seems to be the Gysin sequence of the double cover $\pi:{\rm Gr}^+(k,n)\to {\rm Gr}(k,n)$, which (omitting $\mathbb{Z}_2$ coefficients) looks like $$ \cdots \to H^i({\rm Gr}(k,n)) \stackrel{\cup w_1}{\to} H^{i+1}({\rm Gr}(k,n))\stackrel{\pi^*}{\to} H^{i+1}({\rm Gr}^+(k,n)) \to H^{i+1}({\rm Gr}(k,n))\to \cdots $$ and implies that $w_i(\gamma)$ is nonzero in $H^i({\rm Gr}^+(k,n))$ if and only if it's not in the ideal generated by $w_1(\gamma)$. But I guess determining this is not always straightforward. At least it seems from the above sources that $w_2$ is always nonzero when $k=2$ and that $w_2$ and $w_3$ are usually nonzero when $k=3$.

This is a partial answer to the second question.

In the question, I intended $\gamma$ to denote the tautological bundle over $\operatorname{Gr}^+(k, n)$, whereas in Mark Grant's answer, he uses $\gamma$ to denote the tautological bundle over $\operatorname{Gr}(k, n)$. To make things absolutely clear, let $\gamma^+$ denote the tautological bundle over $\operatorname{Gr}^+(k, n)$ instead. As $\pi^*\gamma \cong \gamma^+$, where $\pi : \operatorname{Gr}^+(k, n) \to \operatorname{Gr}(k, n)$ is the natural double cover, we see that $w_i(\gamma^+) = w_i(\pi^*\gamma) = \pi^*w_i(\gamma)$ is non-zero if and only if $w_i(\gamma)$ is not in the ideal generated by $w_1(\gamma)$.

As the ideal $(\overline{w}_{n-k+1}, \dots, \overline{w}_n)$ is generated by elements of degree at least $n - k + 1$, the elements $w_{i_1}(\gamma)\dots w_{i_t}(\gamma)$ with $i_1 + \dots + i_t \leq n - k$ are independent. In particular, for $1 < i \leq \min\{k, n-k\}$, $w_i(\gamma)$ does not belong to the ideal generated by $w_1(\gamma)$ and hence $w_i(\gamma^+) \neq 0$. Therefore, if $k \leq n - k$, then $w_2(\gamma^+), \dots, w_k(\gamma^+) \in H^*(\operatorname{Gr}^+(k, n); \mathbb{Z}_2)$ are all non-zero.

This explains the observation at the end of Mark Grant's answer: $w_2(\gamma^+)$ is non-zero in $H^*(\operatorname{Gr}^+(2, n); \mathbb{Z}_2)$ for $n \geq 4$, and $w_2(\gamma^+), w_3(\gamma^+)$ are non-zero in $H^*(\operatorname{Gr}^+(3, n); \mathbb{Z}_2)$ for $n \geq 6$; in addition, $w_2(\gamma^+)$ is non-zero in $H^*(\operatorname{Gr}^+(3, 5); \mathbb{Z}_2)$.