For example, Walter Lewin says in many lectures that we will never find a magnetic field $B\propto \frac 1 {r^2}$ - why is this?
I believe it must be related to $\nabla \times E= -\partial_t B$, but I don't see why this would make the previous impossible.
A magnetic field of the form
$$ \boldsymbol{B} \propto \frac{\boldsymbol{\hat{r}}}{r^2} $$
is impossible because
$$ \nabla \cdot \left( \frac{\boldsymbol{\hat{r}}}{r^2} \right) = 4 \pi \delta(\boldsymbol{r}), $$
so a magnetic field of this form would violate Maxwell's equations, one of which is
$$ \nabla \cdot \boldsymbol{B} = 0. $$
It seems that a magnetic monopole might produce a magnetic field like this, but magnetic monopoles are forbidden in classical electromagnetism.
I believe what he is implying is that there are no magnetic monopoles (that we know of), at least in classical electrodynamics. A magnet has a south and a north pole (a dipole), which produces a field (the vector potential) $$ \mathbf{A} (\mathbf{r}) = \frac{\mu_0}{4\pi r^2} \frac{\mathbf{m} \times \mathbf{r}}{r} = \frac{\mu_0}{4\pi} \frac{\mathbf{m} \times \mathbf{r}}{r^3} $$ where $\mathbf{m}$ (vectorial quantities are bolded) is the magnetic moment of this N-S-dipole that is kept constant while the source shrinks to a point. This is the limit of a dipole field. You can read more here on Wikipedia.
