We show that discs accreting on to the magnetosphere of a rotating star can end up in a trapped state, in which the inner edge of the disc stays near the corotation radius, even at low and varying accretion rates. The accretion in these trapped states can be steady or cyclic; we explore these states over a wide range of parameter space. We find two distinct regions of instability: one related to the buildup and release of mass in the disc outside corotation, and the other to mass storage within the transition region near corotation. With a set of calculations over long time-scales, we show how trapped states evolve from both non-accreting and fully accreting initial conditions, and also calculate the effects of cyclic accretion on the spin evolution of the star. Observations of cycles such as found here would provide important clues on the physics of magnetospheric accretion. Recent observations of cyclic and other unusual variability in T Tauri stars (EXors) and X-ray binaries are discussed in this context.