IN CALIFORNIA (which is the state you're in, and the only one I am familiar with)
Unlawful Detainers are confidential for the first 60 days. Only the parties to the action, including attorneys of record and anyone that can prove they live in the household can view the record.
Paternity Actions (where the parents were never married) are confidential cases and can only be viewed by the parties to the action -- ever.
Divorces are public record, the case statuses can be viewed online (where those counties have that type of action, LA county does). If you can get the case number (and they are easily obtainable), you can go to the courthouse and view the record, and yes, if you pay $.50 per page, you can get a copy.
Civil and Small Claims are public record from the date they are filed and can be viewed at anytime, by anyone. The same for probate cases, which are currently (in LA County) going through a process where all the documents are being scanned so they can be viewed online.
Juvenile traffic and delinquency cases are confidential and it's against the law to even admit a case exists.