In most places it's still a far stretch but there will be a day that public transport is freely accessible. In some cities it's already happening and for some people in some Western places, but free public transport is still something to work towards.
In Amsterdam there is free public transport for the elderly with low income and little capital. In Eindhoven there's free public transport for everyone older than 65 years regardless of income. In many other Dutch cities experiments have taken place but until now free public transport hasn't materialized yet.
In many countries there have been experiments with free public transport, check freepublictransports.com for a list. In France there are almost 10 cities offering this. In Belgium the story of Hasselt is quite well known.
Black riding
Hopping along with a ticket is possible, but if you get caught you often have to identify yourself and you get fines on your name. If you don't pay those fines you can end up in jail. Likewise if you don't identify yourself.
If you travel without a ticket in other European countries it's often not a big deal. Your name is noted and you'll have to get off at the next stop. Or (for example in the French TGV) you can "buy" a ticket on your name (without spending money) and remain seated till wherever you want to go. You will have to give an address, but no one can check this address. It is not advisable to black ride but it's definitely possible.
We have an extended article about black riding trains on our site if you want to read more about this.
Estonia
Estonia will be the first country with nation-wide free public transport. In the capital Tallinn local residents have been able to take public transport for free since 2013.
https://popupcity.net/estonia-to-become-the-worlds-first-free-public-tra...
Future
Once self-driving cars will be more common it's likely that they will also be part of public transport networks.
The EU keeps decreasing the level of allowed pollution. In February 2018 this has led Germany to start experimenting with free public transport in various cities.
For whoever wants free transport right now, legally, it's still possible to walk, bike or hitchhike.
