Practically non-existent.
Give this map a check. As you can see the two countries with the least worse infrastructure in this regard are Argentina and Brazil; even then it's mostly around Buenos Aires province and São Paulo state, and it's mostly cargo. And at least in Brazil it seems to me it's getting worse over time, as people rely more and more on aeroplanes for long distance travel.
My personal experience fits the above, by the way. I live in Curitiba; it's in a 4M inhabitants metropolitan region, 1300km from Buenos Aires and 400km from São Paulo. And it's a rather important connection between Paraguay (600km to the west, if counting from Ciudad del Este) and the port of Paranaguá (100km to the east).
So, note: metropolis, with important N-S and W-E connections, you'd expect it to be a train hub, right? Well… no. If I were to travel to any of those places I mentioned, there's no reasonable way I'd take a train:
- Sampa - Bus or plane.
- BBAA - Plane. Or bus, if I was a masochist. Because seriously.
- CdE - Bus. I think planes might be an option too, but last time I was there it was by bus.
- Paranaguá - well, there's a touristic train line, so... yay? Expensive as fuck. Car or bus.
So not even in the conditions you'd expect trains the most you'll find them. (inb4 metropolitan trains became a political joke here, although it's for another matter.)