Court with a Capital C
Here's another tip from Bryan A. Garner's The Winning Brief: 100 Tips for Persuasive Briefing in Trial and Appellate Courts.
You should always capitalize 'court', when referring to:
The United States Supreme Court.
The highest court in the jurisdiction in which a filing is made.
The court to which a filing is addressed.
Local rules will sometimes have additional rules about when to capitalize court . For example, the New York Law Reports Style Manual Rule 10.1(h) directs us to, "[c]apitalize the word 'court' when standing alone only when referring to the Supreme Court of the United States, the New York Court of Appeals or the Appellate Division of the Supreme Court."