Who killed Jimmy?

Step by step: Alice is saying: "She had black hair." Remark that only SHE has black hair, and in the note it's said: "that no one accuses themselves". So she's lying. Andrew is saying: "Alice is lying! She had brown hair." Ok, nothing wrong here, Alice is lying and Anna have brown hair, so it's possible. Anna is saying: "I speak the truth! He had brown hair." Well "HE" must be Andrew but he has black hair. So she's lying as well, Andrew is telling the truth and the murderer is probably Anna. Note: The question is asking for who is telling the truth, not who is the murderer.

If I read it correctly, though it maybe implied, the question doesn't explicitly specify that one of the three suspects is the guilty party. Any one of the three could be telling the truth if they are refering to a guilty 4th party.

To those saying Alice could be telling the truth, no, that is not possible. " She had black hair" cannot be true, since "she" cannot refer to Andrew, and "black hair" cannot refer to Anna, and the speaker never refers to him/herself. So the question is fine as stated, with a unique answer.

Presume that the first statement is true -- the suspect is a female with a black hair. However, the condition says that none of them accused themselves, so the first statement isn't true.

Presume that the second statement is true -- the first statement's a lie and that the female suspect had brown hair. This is a potential statement that can tell the only truth.

Presume that the third statement is true -- the suspect is a male with a brown hair. However, the only male suspect's hair color is black, not brown, therefore making her statement a lie.

Therefore, only ANDREW told the truth in this particular case.

I originally wrote this problem more or less two years ago, I left it up here without checking the comments, which was a bad idea because there was a couple problems with it people had that I didn't notice. The original problem was very flawed and thank god a moderator fixed it, this is the fixed version but there may still be some ambiguity in parts for people. Here's a full explanation and solution that hopefully covers some of the problems people might have had:

The only three suspects for the crime are Alice, Andrew and Anna, as stated in the problem. The murderer has to be exactly one of these, as also stated. There is no male with brown hair in this group, so Anna has to be lying. Alice says it was a female with black hair, no one accuses themselves and the only Female with black hair here is Alice meaning she was lying because she was referring to someone not in this group, and at the start it stated the murderer had to be in this group. Alice is indeed lying and there is also a female with brown hair in the group therefore Andrew is the only one telling the truth. We know that none of the people here are transgender because their names tell us their gender identity (All names listed are one gendered, Anna is a girls' name, etc.) and in all cases that identity is the same as the sex they are stated to be.

There is also a possible discrepancy if Andrew's second sentence is meant to say Alice use to have brown hair, rather than stating the killer had black hair at the time of the murder and still does.

Either Andrew or Anna could not tell the truth, since if Andrew/Anna did, Anna/Andrew (respectively) would tell the truth too, that contradicted with given assumption. Hence, Andrew and Anna told the lie, and Alice told the truth, showing that the murder had black hair and was not Alice, that would be Andrew.