This has been requested on English Language Learners (Embed an audio player), English Language & Usage (Why can't we add video (or even audio) clips to posts?) and on Japanese Language (Allow audio uploads).
Short audio clips would be very helpful with pronunciation questions and answers. Many people, while familiar with the International Phonetic Alphabet, can't easily translate from those symbols (ˌɪntə(ː)ˈnæʃənl fəʊˈnɛtɪk ˈælfəbɪt) into sounds or translate something they've heard into IPA.
On English Language & Usage, a question asking for identification of a dialect in a YouTube clip was criticized for forcing readers to go off-site to understand the question. If we had the ability to upload sound clips, the question could possibly have been salvaged.
Another issue with forcing people off-site for audio clips (mentioned in the Japanese Language meta question) is link rot. There's no way to extract the relevant bits of an audio clip and include it in the question or answer. The Help Center explains
Links to external resources are encouraged, but please add context around the link so your fellow users will have some idea what it is and why it’s there. Always quote the most relevant part of an important link, in case the target site is unreachable or goes permanently offline.
Is there some concern about adding support for SoundCloud-hosted audio on language sites that wasn't an issue for the sites where it was enabled? I think for language sites, the clip length could be very limited if storage is the issue.