Three distinct types of high toroidal mode number instabilities are obtained from a comprehensive kinetic calculation, using as input transport code results from the analysis of a recent design for the Burning Plasma Experiment (BPX). These instabilities are: the collisionless trapped electron-ion temperature gradient mode, the magnetohydrodynamic ballooning mode, and a high toroidal mode number version of the 'toroidicity induced Alfvén eigenmode' or 'gap' mode. The dependence of the instability linear eigenfrequencies on minor radius, beta and toroidal mode number is investigated, together with the effects of hot alpha particles. Relative quasi-linear fluxes of particles and energy for each species are also obtained. In addition, the beta dependence of the magnetohydrodynamic ballooning mode is investigated for a case using as input the results of a transport code calculation for the Tokamak Fusion Test Reactor (TFTR) in an extrapolation to a deuterium-tritium mixture. The effects of alpha particles and the relative quasi-linear fluxes are also investigated for this case.