14 pages, 7 figures (reduced resolution).-- Final version of the paper available at: http://prd.aps.org/We examine the linear stability of various configurations in Bose-Einstein condensates with sonic horizons. These configurations are chosen in analogy with gravitational systems with a black hole horizon, a white hole horizon and a combination of both. We discuss the role of different boundary conditions in this stability analysis, paying special attention to their meaning in gravitational terms. We highlight that the stability of a given configuration, not only depends on its specific geometry, but especially on these boundary conditions. Under boundary conditions directly extrapolated from those in standard General Relativity, black hole configurations, white hole configurations and the combination of both into a black hole--white hole configuration are shown to be stable. However, we show that under other (less stringent) boundary conditions, configurations with a single black hole horizon remain stable, whereas white hole and black hole--white hole configurations develop instabilities associated to the presence of the sonic horizons.C.B. has been funded by the Spanish MEC under project FIS2005-05736-C03-01 with a partial FEDER contribution. G.J. was supported by CSIC grants I3PBGP2004 and I3P-BPD2005 of the I3P programme, cofinanced by the European Social Fund, and by the Spanish MEC under project FIS2005-05736-C03-02. L.G. was supported by the Spanish MEC under the same project and FIS2004-01912.Peer reviewed