Joint modelling of longitudinal data and competing risks has grown over the past decade. Despite the recent methodological developments, there are still limited options for fitting these models in standard statistical software programs, which prohibits their adoption by applied biostatisticians. We summarize four published models, each of which has software available for model estimation. Each model features a different hazard function, latent association structure between the submodels, estimation approach and software implementation. Of the four models considered here, the model specifications and association structures are substantially different, thus complicating model-to-model comparison. The models are applied to the ‘Standard and new anti-epileptic drugs’ trial of anti-epileptic drugs to investigate the effect of drug titration on the treatment effects of lamotrigine and carbamazepine on the mode of treatment failure. Notwithstanding the vastly different association structures, we show that the inference from each model is consistent, namely, that there is a beneficial effect of lamotrigine on unacceptable adverse events over carbamazepine and a non-significant effect on the hazard of inadequate seizure control. The association between anti-epileptic drug titration and treatment failure was significant in most models. To allow for the routine adoption of joint modelling of competing risks and longitudinal data in the analysis of clinical data sets, further work is required on the development of model diagnostics to aid model choice.