What kind of writing sample are public policy jobs looking for?

Scour their sites for any papers, reports, fact sheets, explainers etc. they've produced. Read them, a lot, look at the style and tone, and then write some yourself. They don't necessarily have to be about that exact topic (although your interest in the subject is important if you want a job writing and thinking and analyzing policy about it!), but should keep to the "voice" of the various places you're applying. Have someone else proofread your sample. This is really important.

I'm a senior policy advisor in a provincial government. One thing you get asked to do here for policy job interviews almost every time is to write a briefing note. You will be given a news article or sometimes two and asked to read them and write a note that identifies the key issue, provides a summary and context and identifies options or next steps. Or you'll be given a piece of legislation relevant to the ministry and asked to summarize the key points and identify consequences, stakeholder reactions or impacts on existing programs. You are usually given a time limit, which can be a helpful tool for yourself to use as practice in developing the skills to write this kind of material well. The key things are to be concise and clearly state the issue that the document is about. E.g Is there a problem that needs to addressed? is there an issue that needs to be managed? Is a potential policy direction being considered where you are providing advice on the impacts of such an approach? The ability to identify the real issue, state it clearly, and lay out relevant context and options is what we get assessed on - both for interviews and the everyday part of the job. Being able to demonstrate that you can write well and cut to the chase are really important. It's not about writing everything you know on a subject, but demonstrating that you know what's important about an issue (probably obvious advice but worth emphasizing). Both of the previous suggestions about finding good topics and matching tone are good ones if you don't already have something that fits the bill. Good luck!

I would take your best college paper and edit/update it/turn it into a shorter briefing. For some reason, it's hard to do a good job from scratch for a writing sample.

I agree with Yarly--writing a writing sample from scratch is not going to be the best way to show off your skills. It's very difficult to write something like that that does not feel fake or artificial. I recently applied for research and writing position at a major university, and my only job since college had been working in an Information Technology department for six years, so it was a bit of a jump for me to say the least. They required a writing sample, and I did not really have anything I had written professionally, so I took a paper I had written in my senior year of college and re-edited it and used that. I was worried about turning in something so old, but no one in the hiring process batted an eye, and I got the job. So I would not worry so much about how old it is, just look for something that you've written that shows you can think critically and write intelligently. These are qualities that generally don't fade. Don't worry if the topic is very different from the position you're applying for. Good writing speaks for itself.