This thesis is exploring the measurements of the photon polarisation in radiative $B$ decays at LHCb, which are mediated through $b\to s\gamma$ transitions. To ensure optimal physics performance, procedures to align the LHCb detector and to monitor the alignment quality over time are presented. Using data corresponding to an integrated luminosity of $3~\text{fb}^{-1}$, collected in the year of 2011 at the centre-of-energy $\sqrt{s} = 7$~TeV and the year of 2012 at $\sqrt{s} = 8$~TeV in proton-proton collisions, the photon polarisation parameter $A^\Delta$, which is related to the ratio of right- over left-handed photon polarisation amplitudes in $b\to s\gamma$ transitions, is measured by performing an untagged time-dependent analysis of more than $4000$ $B_s^0\to \phi\gamma$ decays. From an unbinned simultaneous fit to the $B_s^0\to\phi\gamma$ and the control channel $B^0\to{K^*}^0\gamma$ data samples, a value of $A^\Delta = -0.98^{~+0.46}_{~-0.52}\text{(stat.)}^{~+0.23}_{~-0.20}\text{(syst.)}$ is measured. This result is compatible with the SM expectation of $A^\Delta_\text{SM} = 0.047^{~+0.029}_{~-0.025}$, within two standard deviations. Furthermore, a study of the feasibility of measuring the photon polarisation in $B^+\to\phi K^+ \gamma$ decays is performed. Background from $B^+\to K^+K^+K^-\pi^0$ decays form the main experimental difficulty for the selection of signal candidates when the calorimetric photons are used for decay reconstruction, but fitting the $\gamma/\pi^0$ separation variable can disentangle effectively the signal from these backgrounds. About $1400$ signal $B^+\to \phi K^+ \gamma$ decay candidates are reconstructed. No clear evidence of resonances in the $\phi K^+$-system is found, thus, the chance of measuring the photon polarisation in this decay is slim. When using converted photons, their good momentum resolution allows the $B$ mass to act as a good discriminating variable to separate signal from $B^+\to K^+K^+K^-\pi^0$ decays. In this case, $53\pm17$ signal candidates are found, where the uncertainty is purely statistical. This result is consistent with the expected number of signal events when correcting the yield obtained from calorimetric photons with the photon conversion rate, the reconstruction efficiency and the $K^+K^-$ mass range. In addition, the direct $CP$ asymmetry in the $B^+\to \phi K^+ \gamma$ decay amplitudes is measured to be $A_{CP} (B^+\to \phi K^+ \gamma) = (1.6\pm2.1\text{(stat.)}\pm0.7\text{(syst.)})\%$. This value is consistent with the SM prediction of $-(0.1-1)\%$.