It's there, though it's far from obvious where to find it. It took me quite some time to find out this info.

After entering your preferred travel details, you get to the dates vs. pricing overview. Select the date (-combination) you want.

Then you get to the web page with the outbound (and inbound if selected) flight options.

Select both an outbound and an inbound option.

Then, the web page does some flashing and at the top, you get an overview of the flight dates and the total costs.
The total costs does have a down arrow next to it at the right side and the total is clickable.
Click on it and you get the detailed overview of all the segments, timings as well as the travel and fare class.

I have to find other locations, where the fare class is being shown
Not even in the "View details" overview.