I'm declaring a field:
private static final String filename = "filename.txt";
First, does the order of private static final matter? If not, is there a standard accepted sequence or convention?
Second, the filename in my application is fixed. Is this the best was to store its value?
No. But that is the sequence I usually see used.
It's a reasonable choice, but some would prefer a configuration file, either Properties or another file format (e.g. XML). That way, you can change the filename without recompiling.
The order doesn't matter, but you can always play around with it - there's only 6 possibilities to test.
I'm not aware of any convention, though I put the visibility modifier first (public/private/protected) so you can eyeball it and it lines up.
If it's fixed then you can do that, but I always think something is a constant only to discover later (during testing, for example) that I want to pass it in. An argument on the command line or a properties file works for that case, and is a minimum of effort to set up.
I use Checkstyle with Eclipse, which results in a warning if the declaration is in a different order to the one you've specified, citing the Java Language Specification (JLS). For example,
private final static String filename = "filename.txt";
results in
'static' modifier out of order with the JLS suggestions.
They have this page which lists the order they expect, though following the links on that page through to the JLS I can't see anything to back up their assertion of a suggested order.
Having said that, the order they suggest seems to correspond to the order in most of the code I've seen, so it seems as good a convention as any to adopt.
It's common in Java to give constants (static final values) an all-uppercase name, so I would write:
private static final String FILENAME = "filename.txt";
See also Code Conventions for the Java Programming Language. (Those are Sun's code conventions that the majority of Java programmers use).