Tradable mobility credit (TMC) schemes are an approach to travel demand management that have received significant attention in recent years. This paper proposes and analyzes alternative market models for a TMC system -- focusing on market design aspects such as allocation/expiration of tokens, rules governing trading, transaction fees, and regulator intervention -- and develops a methodology to explicitly model the dis-aggregate behavior of individuals within the market. Extensive simulation experiments are conducted within a combined mode and departure time context for the morning commute problem to compare the performance of the alternative designs relative to congestion pricing and a no-control scenario. The simulation experiments employ a day-to-day assignment framework wherein transportation demand is modeled using a logit-mixture model with income effects and supply is modeled using a standard bottleneck model. The results indicate that small fixed transaction fees can effectively mitigate undesirable behavior in the market without a significant loss in efficiency (total welfare) whereas proportional transaction fees are less effective both in terms of efficiency and in avoiding undesirable market behavior. Further, an allocation of tokens in continuous time can be beneficial in dealing with non-recurrent events and avoiding concentrated trading activity. In the presence of income effects, despite small fixed transaction fees, the TMC system yields a marginally higher social welfare than congestion pricing while attaining revenue neutrality. Further, it is more robust in the presence of forecasting errors and non-recurrent events due to the adaptiveness of the market. Finally, as expected, the TMC scheme is more equitable (when revenues from congestion pricing are not redistributed) although it is not guaranteed to be Pareto-improving when tokens are distributed equally.