Can a particle detector cross a Cauchy horizon?

Cauchy horizons are well known to exhibit instabilities in classical spacetime dynamics and singularities in quantum field theory. We analyse the response of an Unruh-DeWitt particle detector that falls towards a Cauchy horizon, in terms of the specifics of the horizon, the choice of the quantum state and the specifics of the detector's trajectory. As a prototype, we study in detail the case for the $1+1$ Reissner-Nordstr\"om black hole with a scalar field in the Hartle-Hawking state. Comparisons are made with the response of a detector that falls into a Schwarzschild-like singularity.