Inferring cosmological parameters from time-delay strong lenses requires a significant investment of telescope time; it is therefore tempting to focus on the systems with the brightest sources, the highest image multiplicities and the widest image separations. We investigate if this selection bias can influence the properties of the lenses studied and the cosmological parameters inferred. Using an ellipsoidal power-law deflector population, we build a sample of double- and quadruple-image systems. Assuming reasonable thresholds on image separation and flux, based on current lens monitoring campaigns, we find that the typical density profile slopes of monitorable lenses are significantly shallower than the input ensemble. From a sample of quads, we find that this selection function can introduce a 3.5 per cent bias on the inferred time-delay distances if the properties of the input ensemble are (incorrectly) used as priors on the lens model. This bias remains at the 2.4 per cent level when high-resolution imaging of the quasar host is used to precisely infer the properties of individual lenses. We also investigate if the lines of sight for monitorable strong lenses are biased. The expectation value for the line-of-sight convergence is increased by 0.009 (0.004) for quads (doubles) implying a 0.9 per cent (0.4 per cent) bias on H sub( 0). We therefore conclude that whilst the properties of typical quasar lenses and their lines of sight do deviate from the global population, the total magnitude of this effect is likely to be a subdominant effect for current analyses, but has the potential to be a major systematic for samples of ~25 or more lenses.