  Dispersionless (flat) electronic bands are investigated regarding their
conductance properties. Due to "caging" of carriers these bands are usually
insulating at partial filling, at least on the non-interacting level.
Considering the specific example of a $\mathcal{T}_3$--lattice we study
long-range Coulomb interactions. A non-trivial dependence of the conductivity
on flat band filling is obtained, exhibiting an infinite number of zeros. Near
these zeros, the conductivity rises linearly with carrier density. At densities
half way in between adjacent conductivity-zeros, strongly enhanced conductivity
is predicted, accompanying a solid-solid phase transition.
