  In recent years experiments revealed intriguing hints for new physics (NP) in
$B$ decays involving \bctaunu and $b\to s\ell^+\ell^-$ transitions at the
$4\,\sigma$ and $5\,\sigma$ level, respectively. In addition, there are slight
disagreements in $b\to u \tau\nu$ and $b\to d\mu^+\mu^-$ observables. While not
significant on their own, they point in the same direction. Furthermore,
$V_{us}$ extracted from $\tau$ decays shows a slight tension
($\approx2.5\,\sigma$) with its value determined from CKM unitarity.
Additionally, BELLE found hints for an excess in $B_d\to\tau^+\tau^-$.
Concerning NP explanations, the vector leptoquark $SU(2)$ singlet is of special
interest since it is the only single particle extension of the SM which can (in
principle) address all the anomalies described above. For this purpose, large
couplings to $\tau$ leptons are necessary and loop effects, which we calculate
in this article, become important. Including them in our phenomenological
analysis, we find that neither the tension in $V_{us}$ nor the excess in
$B_d\to\tau^+\tau^-$ can be fully explained without violating bounds from
$K\to\pi\nu\bar\nu$. However, one can account for $b\to c\tau\nu$ and $b\to
u\tau\nu$ data finding intriguing correlations with $B_{q}\to\tau^+\tau^-$ and
$K\to \pi\nu\bar\nu$. Furthermore, the explanation of $b\to c\tau\nu$ predicts
a positive shift in $C_7$ and a negative one in $C_9$, being nicely in
agreement with the global fit to $b\to s\ell^+\ell^-$ data. Finally, we point
out that one can fully account for \bctaunu and $b\to s\ell^+\ell^-$ without
violating bounds from $\tau\to \phi\mu$, $\Upsilon\to\tau\mu$ or $b\to
s\tau\mu$ processes.
