We study the problem of adaptively identifying patient subpopulations that benefit from a given treatment during a confirmatory clinical trial. This type of adaptive clinical trial, often referred to as adaptive enrichment design, has been thoroughly studied in biostatistics with a focus on a limited number of subgroups (typically two) which make up (sub)populations, and a small number of interim analysis points. In this paper, we aim to relax classical restrictions on such designs and investigate how to incorporate ideas from the recent machine learning literature on adaptive and online experimentation to make trials more flexible and efficient. We find that the unique characteristics of the subpopulation selection problem -- most importantly that (i) one is usually interested in finding subpopulations with any treatment benefit (and not necessarily the single subgroup with largest effect) given a limited budget and that (ii) effectiveness only has to be demonstrated across the subpopulation on average -- give rise to interesting challenges and new desiderata when designing algorithmic solutions. Building on these findings, we propose AdaGGI and AdaGCPI, two meta-algorithms for subpopulation construction, which focus on identifying good subgroups and good composite subpopulations, respectively. We empirically investigate their performance across a range of simulation scenarios and derive insights into their (dis)advantages across different settings.[Extended abstract in PDF]