Start by collecting every URL and screenshotting it
Search your full name in quotes, then your case caption, then your name plus the county, and go several result pages deep on each. Every page carrying your case is a separate line on the form and gets judged separately, so a copy you didn't find is a copy Google didn't remove. Copy the exact address out of the browser bar for each one, not the search result link.
Screenshot each page while it's still up, with the URL and the date in the shot. If the site demands money to take the page down, screenshot that demand first, because paying it makes the page vanish along with the only proof Google will accept. You can crop each shot to show only your face, which is what Google's exploitative-sites guidance tells you to do.
If the site charges to take it down, file under exploitative removal practices
File this one through the personal content removal form, under the exploitative removal practices category. All three conditions have to hold: you're a subject of the content in the URL you submit, the website isn't a business review site, and "the website requires payment to the site or other agencies to have the content removed." A fee routed through a third-party reputation service still counts.
Attach the fee screenshots and write the extra note Google asks for on how the website is exploitative: name the amount demanded, where on the site it's demanded, and what happens if you don't pay. Put every URL on that operator's network into the same submission.
Point the form at the personal information, not at the case
The category list is closed: images of minors, child sexual abuse material, personally identifiable info and doxxing content, personal sexual content, and content on sites with exploitative removal practices. There's no category for an accurate arrest record, mugshot, docket or lawsuit, so file on the personal information printed on the page.
The covered list is literal: your address, phone number or email, Social Security or tax ID numbers, bank and credit card numbers, "pictures of your signature or ID," private records like medical records, and confidential usernames and passwords. Doxxing needs a threat attached, or "a significant amount of aggregated personal info without a legitimate purpose." Quote the exact line on the page that carries one of those in the notes field.
Register your details in Results about you and read every denial
Results about you is the self-serve version, and it watches for your phone number, home address, email address, Social Security number, passport number and driver's license number. Register those, then check the queue: requests move through In progress, Approved, Denied or Undone, and a denial arrives by email with a stated reason. Read that reason. It names the route to try next.
Two limits to plan around. Google won't pull results carrying your contact info from pages "owned by Educational or government institutions" or from "Newspapers," so don't file those here. And the monitor only fires on the contact details you registered, so put a monthly reminder in your calendar and re-run the name search yourself.
If you hold a court order, file the legal removal
Report it for legal reasons when a court decision has determined the content is unlawful, or under copyright, trademark or local law. Google's stated position: "determining whether content is illegal is not always a determination that Google is equipped to make, especially without notice from those who are affected." Submit the decision itself, the specific URLs rather than a homepage, and which section of the order compels removal of each URL. An attorney demand letter isn't a mechanism here.
Know the price before you file. Google republishes the notice: "when legally able, we may send a copy of each legal notice we receive to the Lumen project for publication." Your contact information is withheld. The URLs and the claim are not. And the right to be forgotten covers the EU, Turkey, Serbia and Quebec, so a US resident has no route there.
File Refresh Outdated Content only after the page is down
This one is for people who don't own the page, and it fires only where the page "no longer exists or is significantly different." It "will not remove a page or image that still exists and has not changed." Submit a live page and you get a "Content still on page" denial telling you to contact the webmaster and resubmit afterward.
Run it the day after the site takes the page down or edits it, and not before. Statuses are Pending, Approved, Denied, Expired and Cancelled. Pending runs a few days, and an approved refresh goes Expired after 180 days. A verified site owner can cancel your request, so check the status back rather than assuming it held. Skip Search Console's Removals tool entirely: it only works on a property you've verified ownership of.
What's still standing after Google says yes
The page stays live. Bing and DuckDuckGo keep serving it, anyone holding the direct link still opens it, and Search Console's own page says removal doesn't affect other search engines. Removing a Google result also doesn't seal or expunge the court's own record, and a court order does nothing at Google until you file it on the legal removal form yourself.
Re-run the whole search next month, and the month after that. The cost of doing this yourself is the searching, a screenshot of every copy, one URL per line, four tools' worth of statuses to track, and the same search again when a copy comes back under a new brand.