  Propagation of relativistically intense azimuthally or radially polarized
laser pulses (RPP) in underdense plasmas is demonstrated to be unstable, via 3D
particle-in-cell simulation and disregarding the Kerr non-linearity. Strong
pulse filamentation occurs for RPP in transversely uniform plasma with an
increment, $\Gamma$, close to the well-known one depending on acceleration,
$\alpha$, and modulated density gradient length, $L$, as $\Gamma \approx
(\alpha/L)^{1/2}$. In deep plasma channels the instability vanishes. Electron
self-injection and acceleration by the resulting laser pulse wake is explored.
