The Trax light rail has a small "free fare" zone Downtown
I didn't know this either, and they don't market it so much, but in a certain area of the Downtown, trips within the zone, on bus or light rail, are free.
While there is a lot of writing out there about why transit should be free (see The Tyee series from 2007), most communities lack the budget to do this. Tempe's Orbit bus system still doesn't charge, despite the huge drop in funding (sales taxes)--this service is akin to the intra-neighborhood service that I call "tertiary transit" and complements the regional bus and light rail service provided by Valley Metro.
A few cities offer a "Fareless Square" zone, usually downtown, designed to reduce single occupancy vehicle trips and congestion there.
With the negative impact on transit budgets post the 2008 real estate crash, Fareless districts have been dismantled in both Portland and Seattle. They still exist in Pittsburgh, Salt Lake City, and Calgary (called the "7th Avenue Free Fare Zone").
And separately, on the Northside light rail extension to the stadium district in Pittsburgh, the Pittsburgh Steelers football team and a casino agreed to cover the cost of service on the light rail, so that service is free, but technically, not part of the Downtown zone. See "Trips on North Shore T will be free" from the Pittsburgh Post-Gazette. This is for a three-year period.
In Calgary the free zone only includes the light rail service, not bus. This does touch on the equity concern about "fareless squares." You can argue that such service focused on central business districts privileges people who can afford to pay. Bus service is typically consumed by lower income users. But the point of the service is congestion management, not access.
They don't market it. Someone at the library mentioned it to me. But bus stops in the Free Fare Zone do have a sign topper stating that the stop is in the Free Fare Zone. And you can find it on the companion website Let's Ride UTA, where they discuss issues, including a blog discussion back when they looked to end the service.
Images from the UTA website.