Quantitative magnetic resonance imaging (qMRI) is increasingly investigated for use in a variety of clinical tasks from diagnosis, through staging, to treatment monitoring. However, experiment design in qMRI, the identification of the optimal acquisition protocols, has been focused on obtaining the most precise parameter estimations, with no regard for the specific requirements of downstream tasks. Here we propose SCREENER: A general framework for task-specific experiment design in quantitative MRI. SCREENER incorporates a task-specific objective and seeks the optimal protocol with a deep-reinforcement-learning (DRL) based optimization strategy. To illustrate this framework, we employ a task of classifying the inflammation status of bone marrow using diffusion MRI data with intravoxel incoherent motion (IVIM) modelling. Results demonstrate SCREENER outperforms previous ad hoc and optimized protocols under clinical signal-to-noise ratio (SNR) conditions, achieving significant improvement, both in binary classification tasks, e.g. from 67% to 89%, and in a multi-class classification task, from 46% to 59%. Additionally, we show this improvement is robust to the SNR. Lastly, we demonstrate the advantage of DRL-based optimization strategy, enabling zero-shot discovery of near-optimal protocols for a range of SNRs not used in training. In conclusion, SCREENER has the potential to enable wider uptake of qMRI in the clinic.