In large-scale prediction problems, exhaustively following up on all test units is often impractical and inefficient, motivating a selective reporting strategy that fulfills the dual requirements of informativeness and trustworthiness. Within the InfoFCR (Informative prediction with False Coverage Rate control) framework, we propose SCIP (Selective Conformal Inference for Informative Predictions), a procedure built on three key components: (i) an informative set constructor that tailors prediction sets to individual test units according to user-specified informativeness constraints; (ii) a trust score that provides a principled quantification of the trustworthiness of candidate informative sets; and (iii) generalized conformal p-values that are used to perform FCR analysis for selecting the most promising candidates. We establish that SCIP guarantees finite-sample FCR control and is asymptotically anti-conservative, achieving higher statistical power than existing methods. The framework is highly versatile, accommodating a wide range of error metrics across both regression and classification tasks. Extensive numerical experiments on simulated and real data demonstrate the effectiveness of our approach.