1. He works at the Supreme Court.

2. He works at the supreme court.

3. He is a judge of the High Court.
4. He is a judge of the high court.

5. He is a judge of the District Court.
6. He is a judge of the district court.

7. There are several District Courts here.
8. There are several district courts here.
(Do we have to use capital letters or small letters?)

http://grammar.ccc.commnet.edu/grammar/capitals.htm

I would use #1, because to me it automatically implies the US Supreme Court. But if it's referring to another country's high court or supreme court, I'd leave the capitals off unless I included the specific name of the court. (The High Court of Australia, for example.)

In 5 and 6, again, the capitals would be needed if the specific court is named, such as Harris County District Courts.

I wouldn't use 7 unless the specific name is used. I'd use 8 if the specific name isn't used.

Capitalization rules in relation to court names can vary depending on the jurisdiction and context. However, there are general guidelines that can help determine when to use capital or lowercase letters.

1. When referring to a specific court by its official name, such as the Supreme Court or the High Court, it is customary to capitalize both words, regardless of whether they are used in a sentence or not.

Example: "He works at the Supreme Court."

2. However, when using a more generic term like "supreme court" or "high court" to refer to such a court in a more general sense, lowercase letters are typically used.

Example: "He works at the supreme court."

3. For lower-level courts, such as district courts, the capitalization rules are less standardized. In some jurisdictions, the official name of these courts may be capitalized, while in others, they are not.

Example: "He is a judge of the District Court." or "He is a judge of the district court."

4. When discussing multiple district courts in a general sense, lowercase letters are typically used.

Example: "There are several district courts here."

To ensure accuracy, it is advisable to consult relevant style guides or conventions specific to the jurisdiction or publication you are writing for.