Selection algorithms usually score individual items in isolation, and then select the top scoring items. However, often there is an additional diversity objective. Since diversity is a group property, it does not easily jibe with individual item scoring. In this paper, we study set selection queries subject to diversity and group fairness constraints. We develop algorithms for several problem settings with streaming data, where an online decision must be made on each item as it is presented. We show through experiments with real and synthetic data that fairness and diversity can be achieved, usually with modest costs in terms of quality. Our experimental evaluation leads to several important insights in online set selection. We demonstrate that theoretical guarantees on solution quality are conservative in real datasets, and that tuning the length of the score estimation phase leads to an interesting accuracy-efficiency trade-off. Further, we show that if a difference in scores is expected between groups, then these groups must be treated separately during processing. Otherwise, a solution may be derived that meets diversity constraints, but that selects lower-scoring members of disadvantaged groups.