Regression analysis with multivariable survival data requires specification of a model describing the relationship between predictors and some function of the event time distribution. Popular choices include proportional hazards (PH), accelerated failure time (AFT), and additive hazards (AH) models. Each model imposes an a priori assumption that, respectively, hazard ratios, relative time scales, or hazard differences, associated with a given change in a predictor value, are constant during the entire follow-up period. However, the effects of some of the predictors of interest may not be consistent with the underlying modeling assumption, which requires extending the model to include time-dependent effects. In addition, for each continuous covariate a suitable functional form of its relationship with the outcome has to be determined. Several flexible methods for addressing these modeling challenges were proposed in the literature but there is little evidence regarding head-to-head comparisons of flexible extensions of PH vs. AFT vs. AH models in real-world analyses. We first present a brief overview of selected flexible methods available to estimate time-dependent effects and, for continuous variables, non-linear effects. We also identify the software that allows the implementation of such computationally intensive flexible models. The practical importance of these challenges is illustrated using a case study of prognostic factors associated with cancer mortality.