What writing samples are appropriate to apply for this job?

If there's a contact listed for the position, I would go ahead and ask that person. I don't think they'd look negatively upon this, especially if they weren't specific about the content/style/category of the writing samples. Maybe they can even given you a few prompts. Unless they are overwhelmed with applicants, I can't see them refusing to clarify. This is helping them make a good hiring decision.

I'd say if you have any good technical writing style feasibility reports or status reports or that sort of thing, it might be ideal. Instruction sets / manuals would also be good, if you've got any of those.

While there's nothing wrong with one of your samples being academic, professional material would be better -- because it suggests you've continued to develop (and write) in the years since school. Assuming this is not an entry-level job, you don't want to be grouped in with the fresh graduates (who'll be expected to deliver academic-type work) Most employers who ask for a sample are primarily interested in your overall writing ability (does it seem like you could write a professional-sounding email or letter to a client or a donor, or will your boss need to correct your subject-verb agreement before sending anything?). To a certain extent, they're also looking for range. It takes different skills to write documentation, power-point bullet points, a project debrief report, an RFP, or a formal reprimand to a problem employee... and yet, it wouldn't surprise me if a content manager could find him/herself writing any of these. Having a range of writing samples show that you're comfortable with multiple writing styles, and know when to use them appropriately. If the job primarily involves online content, then at least one of your writing samples should be a piece that was intended to be read online. An astute hiring manager will also be evaluating things like your tone and your audience (and how appropriate your tone is for that audience... and whether that audience is similar to the museum's). Depending on the authority of the position, they may also look for an ability to clearly give written directions, and/or document new processes. Have you considered writing a "Howto" article that teaches other people how do do your job as a Web Content Manager? Perhaps a step by step guide for transitioning a web site to a particular CMS?

Give yourself 3 writing assignments. You will earn mega bonus points if you can demonstrate some familiarity with the museum and their collections. Visit the museum, pick three different exhibits/items on display/event and write a 1-2 page essay or editorial about it as if you were trying to get other potential visitors excited. Pretend you are writing an article for a magazine or newspaper. Alternately, pick some historical person, place or thing and write about that. If you are writing web content for a museum, it will need to be concise, engaging and readable by a wide range of english reading age and ability. Avoid big, pretentious sounding words. Have someone proof-read the writing samples before you submit them.