  We explore various notions of stability for surfaces embedded and immersed in
spacetimes and initial data sets. The interest in such surfaces lies in their
potential to go beyond the variational techniques which often underlie the
study of minimal and CMC surfaces. We prove two versions of Christodoulou-Yau
estimate for $\mathbf{H}$-stable surfaces, a Cohn-Vossen type inequality for
non-compact stable marginally outer trapped surface (MOTS), and a global
theorem on the topology of $\mathbf{H}$-stable surfaces. Moreover, we give a
definition of capillary stability for MOTS with boundary. This notion of
stability leads to an area inequality and a local splitting theorem for free
boundary stable MOTS. Finally, we establish an index estimate and a diameter
estimate for free boundary MOTS. These are straightforward generalizations of
Chen-Fraser-Pang and Carlotto-Franz results for free boundary minimal surfaces,
respectively.
