  Given an Orlicz space $ L^2 \subseteq X \subseteq L^1$ on $[0,1]$, with
submultiplicative Young function ${\mathrm{Y}_X}$, we fully characterize the
closed null sets $\Xi$ of the real line with the property that
H\"ormander-Mihlin or Marcinkiewicz multiplier operators $\mathrm{T}_m$ with
singularities on $\Xi$ obey weak-type endpoint modular bounds on $X$ of the
type
  \[
  \left|\left\{x\in \mathbb R : |\mathrm{T}_m f(x)| >\lambda\right\}\right|
\leq C \int_{\mathbb R} \mathrm{Y}_X \left(\frac{|f|}{\lambda}\right), \qquad
\forall \lambda>0.
  \] These sets $\Xi$ are exactly those enjoying a scale invariant version of
Zygmund's $(L\sqrt{\log L},{L^2})$ improving inequality with $X$ in place of
the former space, which is termed multiscale Zygmund property. Our methods
actually yield sparse and quantitative weighted estimates for the Fourier
multipliers $\mathrm{T}_m$ and for the corresponding square functions.
  In particular, our framework covers the case of singular sets $\Xi$ of finite
lacunary order and thus leads to modular and quantitative weighted versions of
the classical endpoint theorems of Tao and Wright for Marcinkiewicz
multipliers. Moreover, we obtain a pointwise sparse bound for the Marcinkiewicz
square function answering a recent conjecture of Lerner. On the other hand,
examples of non-lacunary sets enjoying the multiscale Zygmund property for each
$X=L^p$, $1<p\leq 2$ are also covered.
  The main new ingredient in the proofs is a multi-frequency, multi-scale
projection lemma based on Gabor expansion, and possessing independent interest.
