  Thin sheets deposited on a substrate and interfaces of correlated materials
offer a plethora of routes towards the realization of exotic phases of matter.
In these systems, inversion symmetry is broken which strongly affects the
properties of possible instabilities -- in particular in the superconducting
channel. By combining symmetry and energetic arguments, we derive general and
experimentally accessible selection rules for Cooper instabilities in
noncentrosymmetric systems which yield necessary and sufficient conditions for
spontaneous time-reversal-symmetry breaking at the superconducting transition
and constrain the orientation of the triplet vector. We discuss in detail the
implications for various different materials. For instance, we conclude that
the pairing state in thin layers of Sr$_2$RuO$_4$ must, as opposed to its bulk
superconducting state, preserve time-reversal symmetry with its triplet vector
being parallel to the plane of the system. All pairing states of this system
allowed by the selection rules are predicted to display topological Majorana
modes at dislocations or at the edge of the system. Applying our results to the
LaAlO$_3$/SrTiO$_3$ heterostructures, we find that while the condensates of the
(001) and (110) oriented interfaces must be time-reversal symmetric,
spontaneous time-reversal-symmetry breaking can only occur for the less studied
(111) interface. We also discuss the consequences for thin layers of
URu$_2$Si$_2$ and UPt$_3$ as well as for single-layer FeSe. On a more general
level, our considerations might serve as a design principle in the search for
time-reversal-symmetry-breaking superconductivity in the absence of external
magnetic fields.
