This site lists all the available European government bonds:

https://www.eurex.com/ex-en/markets/int/fix/government-bonds

Can somebody, who trades these regularly, tell me which ones are the most important ones, i.e. the counterparts to the US ZF,ZN,ZB and UB futures, and the main spreads that Eurobond traders are looking at? Is there a list for those spreads similar to what the CME website offers (NOB, BOB..)?

I don't actively trade the European rates products but since no one else responded I'll simply say that there really isn't a true counterpart in Europe to the US rates products. For one thing the US market is way, way bigger and deeper, with trillions of dollars in outstanding issuance, monthly note and bond auctions, and a fully populated yield curve from overnight all the way out to 30 years. None of the individual European markets are deep or broad enough to tranche out and spread the way we can in the US.

All that said, the most active European bond markets are gilts and bunds. OATS, BTPs etc. are third tier. There are liquidity, credit, and in some cases currency issues that must also be considered in any spread relationships, which for a retail or individual trader can be daunting.

Also, you may find more info and/or respondents to your queries in the dedicated Notes and Bonds section of the forum.