"The Adventure of the Blue Carbuncle" can be divided into two distinct parts: in the first, Holmes details for Watson his methods for constructing a profile of Henry Baker, all drawn from the state of his hat. Later, the commissionaire Peterson returns with news that a stolen gem had been found in Baker's goose. Holmes proceeds to conduct an investigation, working backwards to unravel the crime.
I would say that the first part of the story is where we really see the heavy use of clues (in this case, to build a profile for Mr. Baker). From the hat's size, Holmes deduces that Baker must possess a large brain and thus a great deal of intelligence. Given the hat's expensive nature, as well as its current state of decline, he concludes that Baker had once been well off financially but not at present. The hat-securer suggests that Baker had once been a person with foresight, since hat-securers are not sold with these hats but are invaluable in times of bad weather. However, the fact that he had not replaced the broken elastic would suggest that his powers of foresight had declined as well.
Further physical investigation of the hat reveals details about his hair and the cream he uses in it. The hat's dustiness, Holmes claims, suggests marital problems (and the details of the goose, which had been purchased for one Mrs. Henry Baker, rules out his being a bachelor). Thus, Holmes has produced in an extensive profile of the person in question.
From here, the story transitions into an active criminal investigation via the discovery of the stolen gem. Here, Holmes must begin with Baker himself (as it was in his goose that the gem was discovered) and work backwards to discern the nature of the crime. Intending to test Baker's own innocence, he has Peterson advertise the discovery of the hat and goose. When Baker arrives, he returns the hat but reveals that the goose had already been eaten (though Holmes has provided him a substitute). Baker's ready acceptance of that substitute reveals his own innocence, and their interview provides Holmes information from which he can reconstruct this sequence.
From here, Holmes travels to the Alpha Inn, where Baker had received his goose from Mr. Windigate, as part of a goose club. Here, he learns that Windigate himself had received his geese from a seller named Breckinridge. When Breckinridge proves uncooperative, however, Holmes plays to the man's pride and proclivity towards gambling, insisting that the bird he had eaten was country bred, even as Breckinridge tells him otherwise. When Holmes wagers money on this opinion, Breckinridge shows him his books, thus giving Holmes information as to where he had purchased his geese.
After Holmes exits, he spies a new arrival, arguing with Breckinridge, insisting on receiving his goose. This newcomer is revealed to be James Ryder of the Hotel Cosmopolitan. Now Holmes can interview the guilty party.