References for Stiefel-Whitney class of Stiefel manifolds and Grassmannians Let $M$ be a manifold. The total Stiefel-Whitney class of $M$ is defined to be the Stiefel-Whitney class of the tangent bundle $TM$
$$
w(M)=1+w_1(TM)+w_2(TM)+\cdots
$$
I want to find references for
$$
w(SO(n)/SO(k)),(i.e. w(V_{n-k}(\mathbb{R}^n))), \\
w(U(n)/U(k)),(i.e. w(V_{n-k}(\mathbb{C}^n))), \\
w(Sp(n)/Sp(k)),(i.e. w(V_{n-k}(\mathbb{H}^n))),\\
w(SO(n)/(SO(k)\times SO(n-k))),(i.e. w(G_{n-k}(\mathbb{R}^n))), \\
w(U(n)/(U(k)\times U(n-k))),(i.e. w(G_{n-k}(\mathbb{C}^n))), \\
w(Sp(n)/(Sp(k)\times Sp(n-k))),(i.e. w(G_{n-k}(\mathbb{H}^n))).
$$
Which ones of the above are known? Where could I find these formulas?
 A: The Stiefel manifolds are all parallelizable for $n-k\ge2$, so their total Stiefel-Whitney classes are equal to $1$. A reference is Theorem 3.1 of this paper by Kee Yuen Lam.
For the finite Grassmannians, things are a little more complicated. In the complex and symplectic cases you should be able to calculate these using the Wu formula. Partial results on the normal Stiefel-Whitney classes $\bar{w}(G_{n-k}(\mathbb{R}^n))$ are scattered throughout the literature, a recent reference being 
Korbaš, J.; Novotny, P. On the dual Stiefel-Whitney classes of some Grassmann manifolds. Acta Math. Hungar. 123 (2009), no. 4, 319–330.
This article and the articles it references are content with showing non-triviality of some normal SW-class in order to deduce non-immersion results, and do not give formulae for $w(G_{n-k}(\mathbb{R}^n))$.
Most of the methods seem to use the vector bundle isomorphisms
\begin{align*}
T(G_{n-k}(\mathbb{R}^n)) & \cong \operatorname{Hom}(\gamma_k,\gamma_{n-k})\\
                          & \cong \gamma_k^*\otimes \gamma_{n-k} \\
                          & \cong \gamma_k\otimes \gamma_{n-k}\end{align*}
and the splitting principle.
A: In this note, the first two Stiefel-Whitney classes of unoriented, oriented, and complex grassmannians are determined in terms of the Stiefel-Whitney classes of their tautological bundles. This is achieved via the method mentioned at the end of Mark Grant's answer.
For the unoriented grassmannian $\operatorname{Gr}(m, m+n) = O(m+n)/(O(m)\times O(n))$, we have 
\begin{align*}
w_1(\operatorname{Gr}(m, m+n)) &= (m + n)w_1(\gamma)\\
w_2(\operatorname{Gr}(m, m + n)) &= \left[\binom{m}{2} + \binom{n}{2} + m^2 + mn - 1\right]w_1(\gamma)^2 + (m^2 + n^2)w_2(\gamma)\\
&= \begin{cases}
0 & m - n \equiv 2 \bmod 4\\
w_2(\gamma) & m - n \equiv 1 \bmod 4\\
w_1(\gamma)^2 & m - n \equiv 0 \bmod 4\\
w_2(\gamma) + w_1(\gamma)^2 & m - n \equiv 3 \bmod 4.
\end{cases}
\end{align*}
For the oriented grassmannian $\operatorname{Gr}^+(m, m+n) = SO(m+n)/(SO(m)\times SO(n))$, we have 
\begin{align*}
w_1(\operatorname{Gr}^+(m, m + n)) &= 0\\
w_2(\operatorname{Gr}^+(m, m + n)) &= \begin{cases}
0 & m - n \equiv 0 \bmod 2\\
w_2(\gamma_+) & m - n \equiv 1 \bmod 2.
\end{cases}
\end{align*}
For the complex grassmannian $\operatorname{Gr}^{\mathbb{C}}(m, m + n) = U(m + n)/(U(m)\times U(n))$, we have
\begin{align*}
w_1(\operatorname{Gr}^{\mathbb{C}}(m, m + n)) &= 0\\
w_2(\operatorname{Gr}^{\mathbb{C}}(m, m + n)) &= (m + n)w_2(\gamma_{\mathbb{C}}).
\end{align*}
As for the quaternionic grassmanian $\operatorname{Gr}^{\mathbb{H}}(m, m + n) = Sp(m + n)/(Sp(m)\times Sp(n))$, it is $2$-connected, so $w_1(\operatorname{Gr}^{\mathbb{H}}(m, m + n)) = 0$ and $w_2(\operatorname{Gr}^{\mathbb{H}}(m, m + n)) = 0$.
In principle, you can calculate all of the Stiefel-Whitney classes using the splitting principle argument, but the calculations get more and more complicated. A short cut for the third Stiefel-Whitney class is to use the fact that $\operatorname{Sq}^1(w_2) = w_3 + w_1w_2$. It follows that we have
\begin{align*}
w_3(\operatorname{Gr}(m, m + n)) &= \begin{cases}
0 & m - n \equiv 0, 2 \bmod 4\\
w_3(\gamma) & m - n \equiv 1 \bmod 4\\
w_3(\gamma) + w_1(\gamma)^3 & m - n \equiv 3 \bmod 4
\end{cases}\\
& \\
w_3(\operatorname{Gr}^+(m, m + n)) &= \begin{cases}
0 & m - n \equiv 0 \bmod 2\\
w_3(\gamma_+) & m - n \equiv 1 \bmod 2
\end{cases}\\
& \\
w_3(\operatorname{Gr}^{\mathbb{C}}) &= 0\\
& \\
w_3(\operatorname{Gr}^{\mathbb{H}}) &= 0.
\end{align*}
