I would say, this depends on what you want/need to observe.
As a matter of fact, real-time monitoring in the debugger is usually not possible.
Thus a scope or logic analyzer is recommended. Even the cheaper ones' supported by sigrok work quite well.
Consider transmission errors, and the resulting Rx error flags.
I usually catch all receive errors in the associated interrupt handler, and often ignore them henceforth. Which means, the flag must usually be cleared in the handler for the UART to continue to work (check the reference manual !), but there is no code to treat them specifically.
To "real-time debug" certain Tx / Rx routines associated with a limited amount of data, you could store associated data and/or states in an instrumentation buffer, set a breakpoint after the routine finished, and then inspect the contents with the debugger.
Or, using a scope / logic analyzer again, you could toggle GPIOs to monitor specific states.
As a side note, in some cases you can get away for quite some time in the development process with a simulated external device - either by injecting the "receive data" from within instrumentation code, or from an simple ad-hoc PC program.
And of course there is the option of high-value debug adapters (in the 4-digit price range) with separate trace channels, which allow proper real-time data tracing.
The common printf() - style semihosting output does usually not work in this case, and can even mess up the entire application timing.
