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Anyone can easily search for ArcGIS servers.
Assume you want to know if St. Louis County Minnesota (County seat is Duluth) has a public-facing ArcGIS server. First, find the county website. Note that the website address includes “stlouiscountymn”. Second, do a Google search on: stlouiscountymn “rest/services”. Include the quotes in your search.
If this county has a GIS server then you will likely see search hits to various places within the table of contents. Open any table of contents page and then in the upper left corner click on “Home”. You now will see the ‘top’ of the table of contents for this ArcGIS server and the browser address bar will display an address that ends in “/rest/services”.
Of course at the county and city level you could simply call or email the GIS staff and ask. Here is one way to phrase the question: “Do you have a public-facing ArcGIS server and, if so, what is the rest service endpoint?”
Sometimes you might see two server addresses that appear identical except one of them says “arcgis” and the other says “ArcGIS”. Technically these are different internet addresses and these two servers may or may not contain the same information.
Tip: If you are interested in a topic instead of a state/county/city then try a Google search like this example: “climate change” “rest/services”. Include keywords for your topic and always include rest/services in quotes. Many of the hits that are returned will point you to ArcGIS layers.
Some useful information from Joseph Elfelt’s amazing List of ARC GIS Servers.