Let $G$ be a finite infinitesimal group scheme (e.g.$\mu_p,\alpha_p) $ over a perfect field $k$, how much is known about $H^1_{fppf}(k,G)$?

I think that $H^1(k,G)=1$ for all infinitesimal $G$. Let us make some preliminary comments on the process of perfection. If $X=Spec(A)$ is an affine $k$scheme then we can form its perfection $X^{perf}=Spec(A^{perf})$ where $A^{perf}=\varinjlim_{\sigma} A$ is the direct limit of the $\mathbb{N}$indexed system formed by the Frobenius of $A$. There is a canonical map $X^{perf}\to X$. If $X$ is a $k$group scheme, then $X^{perf}$ is a $k$group scheme. If $X$ is a torsor under a $k$group scheme $G$, then $X^{perf}$ is a torsor under the $k$group scheme $G^{perf}$. Let us come back to the question. Let $X$ be a torsor under $G$. Then $X^{perf}$ is a torsor under $G^{perf}$. Since $G$ is infinitesimal we have $G^{perf}=1$. Thus $X^{perf}=Spec(k)$. The map $X^{perf}\to X$ shows that $X$ has a $k$point, hence is trivial. 

