We investigate the effect of black hole spin on warped or misaligned accretion discs - in particular i) whether or not the inner disc edge aligns with the black hole spin and ii) whether the disc can maintain a smooth transition between an aligned inner disc and a misaligned outer disc, known as the Bardeen-Petterson effect. We employ high resolution 3D smoothed particle hydrodynamics simulations of $alpha$-discs subject to Lense-Thirring precession, focussing on the bending wave regime where the disc viscosity is smaller than the aspect ratio $alpha lesssim H/R$. We first address the controversy in the literature regarding possible steady-state oscillations of the tilt close to the black hole. We successfully recover such oscillations in 3D at both small and moderate inclinations ($lesssim 15^{circ}$), provided both Lense-Thirring and Einstein precession are present, sufficient resolution is employed, and provided the disc is not so thick so as to simply accrete misaligned. Second, we find that discs inclined by more than a few degrees in general steepen and break rather than maintain a smooth transition, again in contrast to previous findings, but only once the disc scale height is adequately resolved. Finally, we find that when the disc plane is misaligned to the black hole spin by a large angle, the disc tears into discrete rings which precess effectively independently and cause rapid accretion, consistent with previous findings in the diffusive regime ($alpha gtrsim H/R$). Thus misalignment between the disc and the spin axis of the black hole provides a robust mechanism for growing black holes quickly, regardless of whether the disc is thick or thin.
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