Hey all, Monica speaking. One of the hardest questions to Google when applying to graduate school is how to write a statement of purpose. There is a lot of advice for how to apply to grad school in general, but science graduate applications are a different beast because of how formal they have to be. There's a particular art to selling yourself, in a based-on-my-work way, that's hard to capture.
One resource which I quite like is CMU's advice handout. This does a really good job of capturing what should go into a statement of purpose. This blog post also does a great job of telling you what you SHOULDN'T be writing. The statement of purpose is not a college application essay-- you're selling your ability to do research, not you as a person.
Given how difficult it is to find examples of essays, I've attached my two essays to Berkeley. One is a statement of purpose, which is the usual requirement to schools. Berkeley was the only one of my schools which also required a "personal history" essay-- and I actually think this might have been important in my admission, but at most schools you don't have the opportunity to write a document like this. Regardless, I'm making them both accessible as examples (Statement of Purpose, Personal History). I don't have a good sense of how these are different from other people's application materials, so take them with a grain of salt :).
However, there is good news! Though graduate essays are not often available (it's also considered a bit taboo to ask people for them), there is a marvelous website that collects essays for the NSF GRFP Fellowship. I highly recommend reading the essays under "Personal" on this website put together by Alex Lang (scroll to the bottom where the essay examples are), because these personal statements are almost exactly the style you want to use in your graduate applications. They might be leaning towards the too-personal end, but reading a bunch of these will let you get the gist of what your essays should look like.
I wrote similar essays for each graduate school I applied to, with a few paragraphs and sentences that were specific to each program. Make sure that for your main essay, you have professors-- preferably more than one professor-- read it, and the recommendation is always to start early since you'll likely need to go through several drafts and make major changes. Also have your friends read it, especially if you know people in graduate school. I personally am happy to read essays right now-- send me an email (see About page) and we can Google Doc it out.
Best of luck!