The National Science Foundation's Graduate Research Fellowship Program is a prestigious, national (US-based) award for students purusing PhDs in STEM fields and the social sciences. As of 2019, student can apply once before entering graduate school (i.e. while they are applying to grad programs) and once in either their first or second year of graduate school.
Every eligible PhD student should apply! It is highly competitive and therefore (i) very rewarding if you receive it and (ii) good experience for any future grants and fellowships you apply for. I applied twice: once unsuccesfully in Fall 2017, as I was putting in my grad school applications, and once successfully in Fall 2019. I've put my (successful) application materials here for other applicants to reference. My research proposal is for a synchronic and diachronic study of word order in Valley Zapotec, and so these materials are particularly relevant for students applying to work in language documentation and description.
The primary negative comment from reviewers on my application was that I was vague in explaining (i) my precise methodology and (ii) what the broader outcomes would be and how I would accomplish them.