  We prove multiple vector-valued and mixed-norm estimates for multilinear
operators in $\rr R^d$, more precisely for multilinear operators $T_k$
associated to a symbol singular along a $k$-dimensional space and for
multilinear variants of the Hardy-Littlewood maximal function. When the
dimension $d \geq 2$, the input functions are not necessarily in $L^p(\rr R^d)$
and can instead be elements of mixed-norm spaces $L^{p_1}_{x_1} \ldots
L^{p_d}_{x_d}$.
  Such a result has interesting consequences especially when $L^\infty$ spaces
are involved. Among these, we mention mixed-norm Loomis-Whitney-type
inequalities for singular integrals, as well as the boundedness of multilinear
operators associated to certain rational symbols. We also present examples of
operators that are not susceptible to isotropic rescaling, which only satisfy
``purely mixed-norm estimates" and no classical $L^p$ estimates.
  Relying on previous estimates implied by the helicoidal method, we also prove
(non-mixed-norm) estimates for generic singular Brascamp-Lieb-type
inequalities.
