AbstractWe study invariance for eigenvalues of selfadjoint Sturm–Liouville operators with local point interactions. Such linear transformations are formally defined by or similar expressions with instead of δ. In a probabilistic setting, we show that a point is either an eigenvalue for all ω or only for a set of ω's of measure zero. Using classical oscillation theory it is possible to decide whether the second situation happens. The operators do not need to be measurable or ergodic. This generalizes the well known fact that for ergodic operators a point is eigenvalue with probability zero.