The femtosecond GHz-burst mode laser processing has attracted much attention in the last few years. Very recently, the first percussion drilling results obtained in glasses using this new regime were reported. In this study, we present our latest results on top-down drilling in glasses, focusing specifically on the influence of burst duration and shape on the hole drilling rate and the quality of the drilled holes, wherein holes of very high quality with a smooth and glossy inner surface can be obtained. We show that a decreasing energy repartition of the pulses within the burst can increase the drilling rate, but the holes saturate at lower depths and present lower quality than holes drilled with an increasing or flat energy distribution. Moreover, we give an insight into the phenomena that may occur during drilling as a function of the burst shape.