I've been pretty liberal about sharing my PhD statement of purpose with folks who email me, so in the interest of open and equal access, I'm putting them here for anyone to see. I also included some notes on thoughts I have looking back, with some things I would have changed if I were to rewrite it. If you're looking for advice on writing statements of purpose, I like this document by Noah Smith (much of this advice shaped the final document below) and this document by Nathan Schneider. I applied in December 2018 [1]:

PhD Statement of Purpose

I hesitate to do this because it's pretty cringe in hindsight (and my interests have definitely shifted considerably since writing this), but I think it does a reasonable job of avoiding a common failure mode: making your statement of purpose just your CV, but translated into paragraphs. I wrote a fair amount about the past projects that I worked on, but I mainly did so towards the goal of motivating / painting a coherent picture of how I reached my research interests at the time.

For example, the paragraph about my past work at AI2 was mostly to convey that I had hands-on experience with QA models and was pretty disillusioned by the gap between their (hyped) purported capabilities and their actual capabilities.

I also think the paragraph about future career goals could be omitted—don't feel pressured to make any semblance of a decision of what you want to do post-PhD. I thought it might just be useful to mention, since I really enjoyed TAing as an undergrad and one of my letter-writers was someone who I had pretty much only interacted with in the context of TA-ing. You will not be held accountable for whatever career goals you set, people expect them to change :) (and I can confirm that your opinions will certainly develop during your PhD).

If I could change things, I'd definitely write less about my past. For instance, I would shorten the part about the RepL4NLP paper—there's no need to go into the concrete results, it would suffice to just say that my personal experience with the opacity of neural nets led me to do some initial work in the area, and motivated my future interests.

I also regret not being more concrete about maybe specific projects I want to do in the future, but I recognize that it's easy to say that in hindsight. As I was writing the statement, I was definitely afraid that my misinformed senior-year undergraduate research opinions would turn off any NLP faculty with the misfortune of reading my application, so I chose to be conservative instead and say less. Looking back, I think it might have actually looked better to have stronger personal opinions with more evidence for why I feel the way I do.

Have confidence in your ideas and research goals; you've clearly thought them through, so use the statement to convey why you think these problems are interesting, and why your proposed solutions seem reasonable in light of prior work in the area.

[1]: For historical context, BERT was released in October 2018, and BERTology wasn't really a thing yet—anyone writing about analyzing neural nets today probably needs to write more arguing why this is interesting, and what new things they want to bring to the table.