Scandinavia Pass

Copenhagen does have the S-tog system, similar to German S-Bahns. I don't know about Denmark, but railpasses do work on German S-Bahns, since they are operated by Deutsche Bahn. However, even if they do work, it would be better to have a local transit card so you can use the Metro, buses and water buses to get around the city.

I assume you're asking about the Eurail Scandinavia Pass?

Eurail passes are valid on the S-trains (operated by DSB, similar to the DB operated S-Bahn systems). As Sam said, Eurail passes are not valid on buses and the metro.

The Copenhagen Card is a tourist pass that combines public transportation with entrance to many attractions and museums. If you're visiting a lot of attractions the Copenhagen Card can be of good value. If you just want a public transportation pass you get far better value with a City Pass, or even regular zone tickets. You can read more about public transportation and tickets in and around Copenhagen here: https://dinoffentligetransport.dk/tourist/

If you're already using a rail pass travel day to get outside the city, then maybe you'd use that S-train coverage for part of your connection. But you wouldn't normally add extra days (at $25-30 each) on the Eurail pass for that purpose alone.

Thank you guys so much! The information was very helpful! God bless and have an awesome day!
Sherry