Knowledge of the underlying physical mechanisms responsible for the triggering of high-frequency transverse combustion dynamics is of fundamental importance in the development of heavy-duty gas turbine combustors, aircraft engine afterburners, and bipropellant liquid rocket engines. Detailed information about three-dimensional thermoacoustic interactions and local flame dynamics, however, remains largely unknown and unanticipated, mainly because high-amplitude transverse mode instabilities are challenging to excite and detect in well-controlled sub-scale laboratory environments. To overcome this impasse, here we exploit a spatially tailored rectangular injector assembly consisting of ten equidistant horizontal slit nozzles to eliminate the complications of out-of-plane flame dynamics characterization. A total of 56 datasets of self-induced instabilities were acquired over a wide range of operating conditions to understand spatiotemporal phase dynamics and important mode shapes, in conjunction with 2D Rayleigh angle reconstruction and phase-resolved OH PLIF-based local flame front identification. Experimentally, we show that high-frequency transverse instabilities are excited only under high temperature and high thermal power conditions, manifested as non-evanescent pressure fluctuations at 6.50 kHz strongly coupled to the second-order tangential mode of the rectangular combustion chamber. Two vertically-oriented pressure nodal planes and the characteristic phase transition perpendicular to the horizontal slit injector direction are accurately measured and reconfirmed by Helmholtz simulations in terms of their interpositions and spatial orientation. Remarkably, the periodic formation of co-propagating coherent structures and concomitant local flame displacement/pinch-off are revealed to play an important role in driving the high-frequency hydrogen combustion dynamics.