In "The Red-Headed League", what are some important clues from the story that help Holmes solve the crime?
The first thing that arouses Holmes's suspicions is in hearing that Mr Jabez Wilson's new assistant in his shop, Vincent Spaulding, was perfectly willing to work for half the usual wage. This sets Holmes's focus upon him from the beginning. It is also Spaulding who first attracted Wilson's attention to the whole extraordinary business of the Red-Headed League. Furthermore, on learning that Spaulding habitually spends most of his waking time in the basement, claiming that he uses it as a darkroom for his photography, Holmes wonders what his real motive is in spending so much time down there. Therefore, Holmes makes a point of setting himself upon Spaulding's track and discovers that he really is John Clay, a known and crafty criminal, and that he has been secretly tunnelling his way from Wilson's basement to the bank. Spaulding is the key to the whole case in this story, and once Holmes fixes on him, it does not take him long to solve the whole mystery.
Secondly, the utterly bizarre nature of the Red-Headed League itself arouses suspicions. On learning that Wilson normally doesn't venture outside, Holmes realises that the whole strange affair has been devised by Clay and his confederates simply to get Wilson out of his shop, on the pretext of easy employment. It is rather an extreme way to go about it, but as Holmes frequently remarks in these stories, the more bizarre cases are generally easier to solve than commonplace ones.
This story is one of the more light-hearted ones in the Holmes canon. Jabez Wilson, with his shock flaming red hair and peculiar tale, comes across as a rather comic figure; indeed, at one point both Holmes and Watson laugh unceremoniously at his expense. Wilson is vexed at this but Holmes pacifies him by remarking on how much the case appeals:
I really wouldn't miss your case for the world. It is most refreshingly unusual.