In this essay we discuss the possibilities and associated challenges concerning beyond the Standard Model searches at FCC-ee, such as rare decays of heavy-flavoured particles and long-lived particles. The Standard Model contains several suppression mechanisms, which cause a given group of processes to happen rarely, resulting in rare decays. The interest in these decays lies in the fact that the physics beyond the Standard Model does not need to be affected by the same suppression mechanism and therefore can naturally manifest in these decays. Their interest is reinforced by the recent report of several measurements of $b$-flavoured rare decays, showing deviations with respect to the Standard Model predictions. We will show how the FCC-ee project has unique capabilities to address these scientific questions and will consider the related detector design challenges to meet. Another group of processes discussed are those that produce new particles with relatively long lifetimes, that travel substantial distances inside the detectors before decaying. Models containing long-lived particles can give answers to many open questions of the Standard Model, such as the nature of dark matter, or the neutrino masses, among others; while providing an interesting experimental complement to mainstream searches. Long-lived particles often display unique experimental signatures, such as displaced tracks and vertices, disappearing tracks, or anomalously charged jets. Due to this, they are affected by very low background levels but in exchange, they often require dedicated reconstruction algorithms and triggers.