Possible spoilers can be found below - consider yourself warned!
Those hoping that DC and Warner Bros are going to make good on a truly connected movie universe are looking to Suicide Squad, which is scheduled to be released only a few months after Batman vs Superman.
There has been plenty of rumors and leaks about how Batman vs Superman is a fairly direct sequel to Man of Steel, with BvS picking up the story in Metropolis after the epic battle with Zod and dealing all of the repercussions that entails. Already there are inquiring minds looking to deduce exactly how Suicide Squad might be picking up the story after Batman vs Superman.
The primary connective tissue between Batman vs Superman and Suicide Squad is the Joker. While this character doesn't appear in Batman vs Superman (Jared Leto was cast in the role after the movie wrapped production), those who have seen the script do report that he is a known quantity, even to Superman.
As production has gotten underway for Suicide Squad, the movie has proved to be not quite as tight a ship as the Batman vs Superman production was and remains to be. Full plot descriptions and scene breakdowns have appeared and even though all of this is of course the stuff of rumor - it has painted a pretty vivid picture.
The Joker is moved from Arkham to the Suicide Squad's base of operations, the prison Belle Reve, as leverage to use against another inmate, Harley Quinn. Why Joker is in prison to begin with is a item of much speculation. Having read a report that the shooting script for Batman vs Superman features a Jim Gordon that is already dead and buried, one could easily deduce that one of the crimes that put Joker behind bars involved the death of Gordon.
The Killing Rumor
A newly released wrinkle in the Suicide Squad rumor mill doesn't necessarily contradict this theory but it does lean in another direction. This Joker-centric information that has been released describes the character in Suicide Squad as having "killed one of Batman's sidekicks" and that there are many references to his past crimes in the movie. Also, he gets to hang out with the psychiatrist of Belle Reve who, wouldn't you know it, is none other than noted psychopath Hugo Strange.
The Hugo Strange rumor is great news and making him a Belle Reve or Arkham psychiatrist is a classic way to introduce his character - but let's unwrap the "killed one of Batman's sidekicks" candy. Obviously, this would lead everyone to believe that the Joker has killed Robin. And it's not like there isn't a whole lot of pages of Batman comics to back this up.
A Death in the Family
The Joker famously killed Jason Todd back in 1988. But like a lot of comic book deaths it didn't end up sticking and came back as a number of different vigilante characters, some more upstanding than others.
What would be interesting is if Batman vs Superman does indeed feature Jena Malone as the female Robin, Carrie Kelley, from the closest thing this movie has to source material, The Dark Knight Returns. Batman taking an assist from Kelley could lead to some backstory as to what happened to other, previous Robins.
We'll see how all of this Robin/Gordon business unravels and what implications it might have as to why the Joker starts out behind bars, but no matter who it is exactly, the fact that it is a "sidekick" of Batman's does mean that when the Joker makes his rumored escape from Belle Reve at the end of Suicide Squad his freedom will be an even bigger thorn in Batman's side. And no I don't mean to disparage the legacy of Jim Gordon by considering him to possibly fall into the "sidekick" category when it comes to this rumor.