In many practical parameter estimation problems, such as coefficient estimation of polynomial regression and direction-of-arrival (DOA) estimation, the exact model is unknown, and a model selection stage is performed prior to estimation. This data-based model selection stage affects the subsequent estimation, e.g., by introducing a selection bias. Thus, new methodologies are needed for both frequentist and Bayesian estimation.  In this study, the problem of estimating unknown parameters after a data-based model selection stage is considered. In the considered setup, the selection of a model is equivalent to the recovery of the deterministic support of the unknown parameter vector. We assume that the data-based model selection criterion is given and analyze the consequent Bayesian and frequentist estimation properties for this specific criterion.  For Bayesian parameter estimation after model selection, we develop the selective Bayesian Cramér-Rao bound (CRB) on the mean-squared-error (MSE) of coherent estimators that force unselected parameters to zero. Similarly, for the frequentist (non-Bayesian) estimation of deterministic unknown parameters, we derive the corresponding frequentist CRB on the MSE of any coherent estimator, which is also Lehmann-unbiased. To this end, the relevant Lehmann-unbiasedness is defined, with respect to the model selection rule.  We analyze the properties of the proposed selective CRBs, including the order relation with the oracle CRBs that assume knowledge of the model. The selective CRBs are evaluated in simulations and are shown as an informative lower bound on the performance of practical coherent estimators. As time permits, I will discuss similar ideas that can be applied to estimation in Good-Turing models.