Given the mess that the US agency org chart is today, I suspect you'd find an equally confusing mess grow out to "deal with" your powered beings.( I'm going to call them mutants for this post, though that's a terrible mis-use of the word. But it is a useful shorthand.)
Stage one: apparent chaos
At first, each department within the federal executive branch would try to cover the aspects of the "strange events" and the "mutants" that most obviously dove-tailed into their existing bailiwick. FBI would develop some teams to deal with the investigation of strange crimes. Secret Service would develop some teams specialized in both the possibility of attacks on the White House and, possibly, risks of supernatural monetary crimes. CIA and NSA would both build out teams to do deal with supernatural espionage. Each military branch would spin up at least some sort of special forces group or tactics for dealing with the threats. (So too would local SWAT teams...) And etc.
During this stage, there would also be distrust of the mutants. There would be scientific investigations. There would also be at least a beginning of recruitment efforts, to bring the "good mutants" into the folds of government to help make sense of and control the "bad mutants."
At some point, a turf war would emerge behind the scenes. A competent executive branch would most likely decide this wasn't solving problems but creating them. This would lead to...
Stage two: Federal Bureau of Supernatural
That's a branstormed title, not necessarily the best title. But the executive branch would try to consolidate the stage one spaghetti into a singular branch that was primarily responsible for monitoring, policing, and generally "dealing with" this new menace. This would draw from the various ad-hoc teams formed by the other divisions. But those other 3-letter branches of government wouldn't entirely shut down their internal teams. They'd be smaller, but not gone. So the FBI would now have fewer teams investigating these events, but they'd still have a few.
Because of that, red tape would still exist as friction between all of the branches. Just like we see today with branches that struggle to figure out who's responsible for various crimes and cases in the real world.
Especially at this stage, you'd see government actively recruiting these special people when doing so gives them an edge. The new division would include the highest ratio of mutants relative to the other, older, branches.
Military might at least toy with the idea of recruiting some kind of "team" or division or units of mutants. Though they may refuse to allow them entry at all. See their history of distrust towards people who are different from the "norm" (Women, people of color, gays, etc.).
Stage three: Hopefully they don't go there
There's a possibility of a third level stage. If the public sees these mutants as a threat, or if they do some horrible, public, thing(s), then pressure might form to suppress these new folks. If even a hint of hatred crops up in civilian populations, at least a few politicians will take advantage of it.
It is possible that entire religions -- or at least sects within religions -- will label these mutants as "demonic" or some variant thereof. This fits to historical precedent, where intolerance is not foreign to some religious sects/religious leaders.
If that hatred rises to some critical mass, then you will get some kind of effort to contain, suppress, deport, or otherwise "deal with the mutant menace." See the X-Men Days of Future Past story arc for how this might go down. Or the real-world Japanese internment camps of WW2. The higher the number of mutants you have, the more likely this becomes, at least to a point.
If this happens, then the FBS or whatever they call themselves, will militarize to a point where they feel they can take on the supers. And smart mutants will try to hide their natures, if possible. This stage may happen in place of stage 2. If the US goes this route, recruiting of mutants will be secretive and publicly disavowed. Mutants may try to seek refuge in foreign countries, to escape the hatred.
External to the US...
Depending on how many of these mutants there are, you'll see some countries welcoming mutants. And you'll see some that hate mutants, banning them and subjecting them to death sentences merely for existing. This will reflect the openness of their cultures to "outsiders" in general. Nations that generally hate differences or "otherness" will equally hate these new mutants.