You didn't really specify what you meant by clean URLs but, maybe this will be helpful to you or someone else.

I have a static site served via nginx, but I don't like seeing .html in the URLs.

It's a little tricky serving a static site without the .html extension because you may have a directory and an html page with the same name.

The way to deal with that is to actually use the .html extension in the file system but not URLs.

  location / {
    if ($uri ~ ^/google) {
      break;
    }
    if ($uri ~ ^/y_key_) {
      break;
    }
    if ($uri ~ \.html$) {
      return 404;
    }
    if ($uri = /index) {
      return 404;
    }
    if (!-f $request_filename) {
      rewrite ^/$ /index.html break;
      rewrite .* $uri.html break;
    }
  }