Lean premixed flames are being used in gas turbines to reduce the emission of NOx. Unfortunately, it results often in thermoacoustic instabilities. These instabilities are high amplitude pressure oscillations which cause damage to the combustion system, reduce efficiency, flame blow-off or flashback. While the cause for instabilities due to longitudinal modes is known, the cause for transverse thermoacoustic instabilities is still not fully understood. Currently, transverse thermoacoustic instabilities are avoided by damping devices that consume considerate amounts of cooling air, thus raising the flame temperature and NOx emissions. Therefore, a good understanding of the feedback cycle that cause transverse thermoacoustic instabilities is crucial in predicting and avoiding combustion instabilities. In this work, Large Eddy Simulations (LES) is used to study processes involved by the driving of instabilities in a small scale burner. Additionally, different dampening alternatives are designed and validated. Experiments have shown that a small scale burner becomes unstable with decreasing flame lengths, due to the addition of hydrogen to the fuel. To research the impact compact flames have on combustion instabilities, simulations of methane flames with decreasing lengths is performed. It is found that decreasing the flame length, increased the amplitude of the pressure oscillations. Similar as to the experiment, both longitudinal, as well as transverse instabilities, is observed in the LES. Unfortunately, to study the transverse thermoacoustic instabilities, the longitudinal modes must be avoided. Hence a modification is designed to dampen the longitudinal mode without affecting the transverse modes. Ultimately an orifice plate was designed that reduced the pressure amplitude of the longitudinal mode by 92 %, while hardly influencing the transverse modes. The sufficient suppression of the longitudinal modes enabled the research into the transverse thermoacoustic instabilities feedback cycle. The research found two driving mechanisms for transverse instabilities. First of all, flames located at pressure anti-nodes are excited by pressure-induced flow oscillations from the premixing ducts. Consequently, the flame’s heat release positively coupled with the acoustic pressure. Additionally, the lateral movement of the flame towards regions of higher pressure, due to transverse acoustic velocity, is observed. Especially at pressure nodes, where acoustic velocities are most potent, the flame movement was substantial. While both mechanisms are found and agree to the driving criteria, it is difficult to say what their actual contribution to the instabilities are. Finally, the actual driving of transverse thermoacoustic instability by flow oscillations inside the premixing duct is studied. At first, a modification is designed to deliberately cause a quarter-wave resonance that forces the pressure anti-node away from the flames. This way, the unsteady heat release is not able to couple with acoustic pressure oscillations. Secondly, a modification inspired by a muffler is aimed at dampening all acoustic waves travelling through the premixing duct. Unfortunately, the quarter-wave resonance created new unstable modes as well as increased the strength of the other instabilities. Also, the muffler modification is not able to dampen the transverse instabilities. Moreover, a resonant mode arose inside one of the muffler elements causing a new instability inside the combustion chamber.