  We use conformal geometry methods and the construction of Friedrich's
cylinder at spatial infinity to study the propagation of spin-$0$ fields
(solutions to the wave equation) on $n$-dimensional Minkowski spacetimes in a
neighbourhood of spatial and null infinity. We obtain formal solutions written
in terms of series expansions close to spatial and null infinity and use them
to compute non-trivial asymptotic spin-$0$ charges. It is shown that if one
considers the most general initial data within the class considered in this
paper, the expansion is polyhomogeneous and hence of restricted regularity at
null infinity. Furthermore, we derive the conditions on the initial data needed
to obtain regular solutions and well-defined limits for the asymptotic charges
at the critical sets where null infinity and spatial infinity meet. In four
dimensions, we find that there are infinitely many well-defined asymptotic
charges at the critical sets, while for higher dimensions there is only a
finite number of non-trivial asymptotic charges that remain regular at the
critical sets.
