How many syllables does Chinese have?
I took the CEDICT file and wrote a script on it. The file has 113k dictionary entries, so it covers a very large portion of the Chinese vocabulary. 

There are 1522 different pinyin syllables in CEDICT, when you distinguish tone numbers (like, ma1, ma2, ma3, ma4, ma). 

If you do not care about tone numbers, you'll get 413 syllables (ma, mo, mi, etc.) Here's a full list alphabetically sorted. 

These results include m2 and m4 (呒, 呣 [interjections]).

UPDATE: Of course, you can argue whether 'ging' and 'fiao' are Mandarin pinyin syllables or not, but they are in the CEDICT file and my answered is based on that.