Important Note: Clyp seems to have changed its API-related policies recently in a way that breaks the routine given below. I do not know how to fix this. I can only guarantee that it was working well at the time it was written.
It took a while, but using ideas from Zach's answer here, I was able to cobble something together:
ClypExport[snd : (_Sound | _Audio), name_String, detailed : (True | False) : False] :=
Module[{raw},
raw = URLExecute["https://upload.clyp.it/upload", {}, "RawJSON",
Method -> "POST",
"MultipartElements" ->
{{StringTemplate["file\"; filename=\"`name`.mp3"] @
<|"name" -> name|>, "audio/mpeg"} -> ExportString[snd, "MP3"]}];
If[raw === $Failed || ! Lookup[raw, "Successful", False], Return[$Failed]];
If[! detailed, raw @ "Url",
Grid[Transpose[{{"Title:", "URL:", "MP3:", "OGG:"},
Prepend[Hyperlink /@
Lookup[raw, {"Url", "SecureMp3Url", "SecureOggUrl"}],
raw @ "Title"]}], Alignment -> Left]]]
which can either return just the plain URL, or a detailed table showing the corresponding *.mp3 and *.ogg file links.
(If you are going to try this in version 11: it still works even though the Method and "MultipartElements" options are being highlighted as unrecognized. One could rewrite this function using URLFetch[] instead if it bothers you, but I'll leave that as an exercise.)
Examples:
zetaSound = Play[RiemannSiegelZ[2000 t], {t, 0, 3}, PlayRange -> {-20, 20}];
ClypExport[zetaSound, "sound of Riemann zeta function"]
"https://clyp.it/hajdomy5"
(* old documentation example *)
chime = Play[Sin[π t/2] Sum[Piecewise[{{((i - 36)/13)^2 Sin[1000 (i + Sin[i]) t],
i/18 < t < (36 - i)/18}}, 0], {i, 36}] // N,
{t, 0, 2}];
ClypExport[chime, "Sine Chime", True]

If you need to modify the routine for more detailed input or output, you can consult Clyp's API.