In this paper, we show that Hilbert transforms along some curves are bounded on $L^p({\mathbb{R}}^n;X)$ for some $1<p<\infty$ and some UMD spaces $X$. In particular, we prove that Hilbert transforms along some curves are completely $L^p$-bounded in the terminology from operator space theory. Moreover, we obtain the $L^p({\mathbb{R}}^n;X)$-boundedness of anisotropic singular integrals by using the “method of rotations” of Calderón–Zygmund. All these results extend preexisting related ones.