In the context of gravitational lensing, the density profile of lensing galaxies is often considered to be perfectly elliptical. Potential angular structures are generally ignored, except to explain flux ratios of point-like sources (i.e. flux ratio anomalies). Surprisingly, the impact of azimuthal structures on extended images of the source has not been characterised, nor has its impact on theH0inference. We address this task by creating mock images of a point source embedded in an extended source and lensed by an elliptical galaxy on which multipolar components are added to emulate boxy or discy isodensity contours. Modelling such images with a density profile free of angular structure allows us to explore the detectability of image deformation induced by the multipoles in the residual frame. Multipole deformations are almost always detectable for our highest signal-to-noise ratio (S/N) mock data. However, the detectability depends on the lens ellipticity and Einstein radius, on the S/N of the data, and on the specific lens modelling strategy. Multipoles also introduce small changes to the time-delays. We therefore quantify how undetected multipoles would impactH0inference. When no multipoles are detected in the residuals, the impact onH0for a given lens is in general less than a few km s−1Mpc−1, but in the worst-case scenario, combining low S/N in the ring and large intrinsic boxyness or discyness, the bias onH0can reach 10−12 km s−1Mpc−1. If we now look at the inference onH0from a population of lensing galaxies with a distribution of multipoles representative of what is found in the light profile of elliptical galaxies, we find a systematic bias onH0of less than 1%. A comparison of our mock systems to the state-of-the-art time-delay lens sample studied by the H0LiCOW and TDCOSMO collaborations indicates that multipoles are currently unlikely to be a source of substantial systematic bias on the inferred value ofH0from time-delay lenses.