Question answering (QA) systems are now available through numerous commercial applications for a wide variety of domains, serving millions of users that interact with them via speech interfaces. However, current benchmarks in QA research do not account for the errors that speech recognition models might introduce, nor do they consider the language variations (dialects) of the users. To address this gap, we augment an existing QA dataset to construct a multi-dialect, spoken QA benchmark on four languages (Arabic, Bengali, English, Kiswahili) with more than 68k audio prompts in 22 dialects from 245 speakers. We provide baseline results showcasing the real-world performance of QA systems and analyze the effect of language variety and other sensitive speaker attributes on downstream performance. Last, we study the fairness of the ASR and QA models with respect to the underlying user populations. The dataset and code for reproducing our experiments are available here: https://github.com/ffaisal93/SD-QA.