"I like sport" is fine. Without an article, 'sport' is uncountable, and it refers to 'sport' as a general concept - the same as saying "I like music". This usage is standard in British English, but not in American English. "I like sports" refers to several individual activities e.g. football, basketball and so on. On its own, "I like a sport" is wrong. We'd only say this as part of a longer sentence e.g. "I like a sport which is quite unusual: it's only played in my country...".

Thank you.

#1 is ungrammatical in American usage. #2 is uncommon because the sport is unusually named (I like soccer). #3 is common.