Is “there goes my chances” or “there go my chances” correct? The first seems to be more natural, but since chances is plural, shouldn’t you use “go” instead of “goes”?
The subject and verb should agree in number. This one is a little tricky as the subject of the sentence is "there", and "there" can be either singular or plural depending on context. You can say, "There is one man in the room" or "There are ten men in the room." Whether you use a singular or plural verb depends on what "there" is referring to.
In this case, as "chances" is plural, the correct verb choice is "go". Here you could also switch both to singular.
There go my chances.
There goes my chance.
Which "sounds more natural" is subjective, I guess. To me, "there goes my chances" sounds horribly wrong. If it doesn't "sound wrong" to you ... well, that's just how it is.
When you get into expressions beginning with there such as there goes my chances or there's five books on the table, you get into an area where strict subject-verb agreement is ignored or redefined by many many native speakers.
I am not going to tell you which one you should say. That is prescriptive grammar. I'm just reporting what people actually say, which is called descriptive grammar. I agree with you that the first one (goes) sounds more natural, as does There's in there's two ways of looking at the issue. See There's vs There are. Note that in this case we are talking about there's, not there is. So even some people who agree with there's two chances I have may not like there goes the two chances I have.