"Apparently" is said of something that appears to be true. It is said of things that you have verified by looking at them - "Apparently, he has not yet eaten breakfast [said when you see that there aren't any breakfast dishes out]"; "Her keys are gone - apparently, she took the car." However, these days you can also use "apparently" for anything that you think is true based on some evidence, even if that evidence is not visual. For example, if a teacher told you that Steve scored poorly on a mathematics exam, even if you didn't see it yourself you could say, "apparently, Steve is not very good at math." You do need some sort of evidence about what you are claiming before you can use "apparently." Most of the time, you should look for visual evidence before using "apparently."
"Seemingly" is said of something that appears to be true, but which is not; or it is said of an appearance which is deceptive. "Sarah's keys are gone. Seemingly, she took the car, but actually, I happen to know that she walked to work today." "He was seemingly happy to have his new job, but in fact he was already looking for different employment." "Seemingly, Steve is having trouble with calculus, but actually he's very good with numbers. The truth is that Steve intentionally failed his math test."
"Evidently" is said of something that is thought to be true because you have some evidence to back it up. It is not very different in use from "apparently," although originally something that was "apparently" true was true because you *saw* some evidence, whereas something that was "evidently" true was likely to be true because of something you had seen, heard, felt, figured out, or been told. Today, the two words are sometimes used interchangeably, although the old distinction is still basically correct. "I looked at the actuary tables, and evidently I'm going to die when I'm 69 years old." "I heard that Steve failed his exam. Evidently, he's no good at math." "Cicero says he believes in the gods, but then he mocks them mercilessly. Evidently, he doesn't really believe in them after all." "Look at all those books! Evidently, Tara really likes reading." "Tom: Where is my orange? Sarah: Evidently somebody took it."