A study of heavy-light meson spectroscopy, specifically the excited and exotic spectra of $B$, $B_s$ and $B_c$ is presented. This work was done on an anisotropic lattice of volume $20^3 \times 128$, with (2+1) flavours of dynamical quarks. A large basis of suitable operators was used in a variational analysis to determine finite volume spectra grouped by lattice irrep for each meson. Spin-identified spectra were produced by assigning continuum quantum numbers $J^P$ to each energy level based on the distribution of dominant operator overlaps, up to spin $J=4$. By examining the operator-state overlaps for each energy level in the lattice irreps, candidate states for a lightest hybrid supermultiplet with $J^P = (0,1,2)^-$ were identified in $B$, $B_s$ and $B_c$.