We present high-precision experiments conducted with the aim to better characterise weak deformations of single cavitation bubbles. Using two needle hydrophones and a high-speed photodetector, we record the timings of shock waves and luminescence emitted at the collapse of laser-induced bubbles and are able to thereby obtain a precise measurement of their displacement during their lifetime. The bubbles are primarily deformed by variable gravity reached aboard parabolic flights, but we additionally take into account the effect of the nearest surfaces. A time shift of approximately 60 ns is found between the bubble lifetimes measured by the hydrophones and the photodetector for spherically collapsing bubbles, which we believe to be a result of different initial shock wave propagation speeds at the bubble's generation and at collapse. The normalised bubble displacement is found to follow a 2/3 scaling law for >0.001, where is the dimensionless anisotropy parameter quantifying the bubble deformation (analogous to Kelvin impulse). Additionally, we quantify the asymmetry of the shock wave generated at the collapse of bubbles with various levels of deformations by comparing the hydrophone signals at two different locations, and find significant variations between the shock peak pressures and energies at >0.001. These results consolidate the suggestion to consider approximate to 0.001 as a practical limit between spherical and deformed bubbles. This limit is probably sensitive to the bubble's initial sphericity, which is exceptionally high in our mirror-based aberration-free setup.