  $B$ decays proceeding via $b\to c\ell\nu$ transitions with $\ell=e$ or $\mu$
are tree-level processes in the Standard Model. They are used to measure the
CKM element $V_{cb}$, as such forming an important ingredient in the
determination of e.g. the unitarity triangle; hence the question to which
extent they can be affected by new physics contributions is important,
specifically given the long-standing tension between $V_{cb}$ determinations
from inclusive and exclusive decays and the significant hints for lepton
flavour universality violation in $b\to c\tau\nu$ and $b\to s\ell\ell$ decays.
We perform a comprehensive model-independent analysis of new physics in $b\to
c\ell\nu$, considering all combinations of scalar, vector and tensor
interactions occuring in single-mediator scenarios. We include for the first
time differential distributions of $B\to D^*\ell\nu$ angular observables for
this purpose. We show that these are valuable in constraining non-standard
interactions. Specifically, the zero-recoil endpoint of the $B\to D\ell\nu$
spectrum is extremely sensitive to scalar currents, while the maximum-recoil
endpoint of the $B\to D^*\ell\nu$ spectrum with transversely polarized $D^*$ is
extremely sensitive to tensor currents. We also quantify the room for $e$-$\mu$
universality violation in $b\to c\ell\nu$ transitions, predicted by some models
suggested to solve the $b\to c\tau\nu$ anomalies, from a global fit to $B\to
D\ell\nu$ and $B\to D^*\ell\nu$ for the first time. Specific new physics
models, corresponding to all possible tree-level mediators, are also discussed.
As a side effect, we present $V_{cb}$ determinations from exclusive $B$ decays,
both with frequentist and Bayesian statistics, leading to compatible results.
The entire numerical analysis is based on open source code, allowing it to be
easily adapted once new data or new form factors become available.
