I think that the main reason many Americans don't trust Hillary are the scandals mentioned by @Brythan in his answer, whether they were exaggerated or not by some sources. But I want to add some additional informations, not about things she did (or didn't do) but about how people view Clinton, and why they don't trust her, i.e. more subjective things based indirectly in what she did or talked about.
Disclaimer: I'm a moderate conservative, and I'm more against Hillary than against Trump (and I'm not an American, if that matters). But my opinions shouldn't matter is this site! So I'll try to be neutral and just show the results of my research.
Before anything, it should be noted that OP's claim that "Hillary is perceived as untrustworthy" is probably true, according to what opinion researches say. This research by Morning Consult shows that "Clinton and Trump Are Historically Unpopular". And the reasons of Hillary's unpopularity are tied to trust. Besides the general "is not trustworthy", many participants of the research said she "is corrupt" and "changes her positions when it's politically convenient" (Donald Trump is seen as "Racist" and Inexperienced").
Now, let's search for some evidences to "why" she is seen as "untrustworthy".
According to this article from left-leaning The Atlantic:
Hillary Clinton can’t be trusted because she’ll do anything to win. That’s what several participants in a focus group of thirty undecided voters moderated by Republican strategist Frank Luntz on Friday in Alexandria, Virginia, seemed to believe. At least some of the group of Democratic, Republican and Independent-leaning voters felt the first woman to win the presidential nomination of a major U.S. political party was too ambitious.
Research indicates that women in leadership positions tend to be evaluated more negatively than men. A 2010 study on backlash against female politicians found that “participants experienced feelings of moral outrage” such as contempt, anger, and disgust when women politicians were described as power-seeking. In contrast, “when participants saw male politicians as power-seeking, they also saw them as having greater agency (i.e., being more assertive, stronger, and tougher) and greater competence.”
It’s difficult to differentiate or untangle lurking sexism from general distrust among voters of the political establishment. Clinton is also a political insider at a time when voters have very little trust in government.
So, according to these researchers, some voters have negative feelings about power-seeking politicians, specially if the politician is a female. However, the article also mentioned the possibility of her position as a political insider to be a source of her being considered untrustworthy for the public.
Another interesting finding is this research:
Research, including new work from our Human Cooperation Laboratory at Yale, suggests Trump may be successful precisely because of his hotheadedness and lack of carefully thought-out proposals. Being seen as calculated can make people trust you.
Hillary Clinton is the opposite of hotheaded. She is careful and calculating – which, despite being a strong asset in actually carrying out the duties of public office, has become a liability in her presidential campaign by undermining the public’s trust in her.
In a recent paper, we found that if you take an action that people like, you come off as much more trustworthy if you decide to act without doing a careful cost-benefit analysis first: Individuals who calculate seem liable to sell out when the price is right.
So, Hillary is seen as "cold and calculated", and people tend to think this kind of people is less trustworthy, according to this research.
There is also this NYT (another left-leaning source) podcast where they discuss exactly this question. According to one of the people they talk, Mark Landler (among others), a "White House correspondent for The Times", about the "high-profile episodes, from Benghazi, Libya, to a private email server, which have contributed to Mrs. Clinton’s troubles":
“Why wasn’t she willing to have a government email account like every other employee of the state department?” Mr. Landler asked on The Run-Up. “That goes to the broader issue around the Clintons: that they simply don’t view themselves as being subject to the same regulations that the rest of the world has to comply with.”
I'll not do the thing Hillary is accused, so I'll not lie: I have not listen the podcast. But maybe someone finds it interesting, so it's here!
Well... Maybe my sources are a bit biased if favour of Hillary (or maybe I'm biased against her so I think my non-biased sources are biased). However, I think these studies can draw a light on the more subjective, unconscious way some people unfavoured Hillary.