  In spacetimes with compact dimensions there exist several black object
solutions including the black-hole and the black-string. These solutions may
become unstable depending on their relative size and the relevant length scale
set by the compact dimensions. The transition between these solutions raises
puzzles and addresses fundamental questions such as topology change,
uniquenesses and cosmic censorship. Here, we consider black strings wrapped
over the compact circle of a $d$-dimensional cylindrical spacetime. We
construct static perturbative non-uniform string solutions around the
instability point of a uniform string. First we compute the instability mass
for a large range of dimensions, $d$, and find that it follows essentially an
exponential law $\gamma^d$, where $\gamma$ is a constant. Then we determine
that there is a critical dimension, $d_*=13$, such that for $d\leq d_*$ the
phase transition between the uniform and the non-uniform strings is of first
order, while for $d>d_*$, it is, surprisingly, of higher order.
