  We study the emergence of string instabilities in $D$ - dimensional black
hole spacetimes (Schwarzschild and Reissner - Nordstr\o m), and De Sitter space
(in static coordinates to allow a better comparison with the black hole case).
We solve the first order string fluctuations around the center of mass motion
at spatial infinity, near the horizon and at the spacetime singularity. We find
that the time components are always well behaved in the three regions and in
the three backgrounds. The radial components are {\it unstable}: imaginary
frequencies develop in the oscillatory modes near the horizon, and the
evolution is like $(\tau-\tau_0)^{-P}$, $(P>0)$, near the spacetime
singularity, $r\to0$, where the world - sheet time $(\tau-\tau_0)\to0$, and the
proper string length grows infinitely. In the Schwarzschild black hole, the
angular components are always well - behaved, while in the Reissner - Nordstr\o
m case they develop instabilities inside the horizon, near $r\to0$ where the
repulsive effects of the charge dominate over those of the mass. In general,
whenever large enough repulsive effects in the gravitational background are
present, string instabilities develop. In De Sitter space, all the spatial
components exhibit instability. The infalling of the string to the black hole
singularity is like the motion of a particle in a potential
$\gamma(\tau-\tau_0)^{-2}$ where $\gamma$ depends on the $D$ spacetime
dimensions and string angular momentum, with $\gamma>0$ for Schwarzschild and
$\gamma<0$ for Reissner - Nordstr\o m black holes. For $(\tau-\tau_0)\to0$ the
string ends trapped by the black hole singularity.
