It's fine, as long as you know what you're doing. I've made several "hot chassis" clocks before, with no problems. That said, you MUST be very careful about what is exposed, so that nobody can get shocked, and be sure to put warning labels on it. Make sure you have properly-sized fuses (ie, smallest possible). I place small-ish series resistors in the power supply to limit the surge current at power-up, and size them to double as fuses if for some reason the actual fuses dont blow (1 one neutral, 1 on hot). It's best to run SPICE simulations on your power supply to balance the inrush current with the wasted energy (heat) of the series resistor.
Half-wave vs full-wave rectification makes a **HUGE** difference. Half-wave is likely to have your circuit ground connected to the neutral wire, which is normally within a few volts of earth ground, hence minimal shock hazard under normal conditions. However, things go wrong and when your life or someone else's is at-risk, you cant take chances.
If you used full-wave rectification, then **EVERY** point in the design is a serious shock hazard. Basically, the (-) lead from the bridge rectifier, which is likely to be your circuit ground, will vary from 0 to minus 310 volts (for 220V mains), or from 0 to minus 180 volts (for 120V mains). I have a few gizmos doing this; again you have to be very careful.
Obviously, connecting anything like a scope will be disastrous unless you have an isolation transformer, because the ground of your scope probe will short-out your hot supply.
But to answer your question, sensitive devices, like raspberry pi computers, are fine in hot-supply systems. Since I use the onboard WiFi, I do all of the software and logging via network, so there is never an external physical connection. As long as everything is at the same potential, there is no hazard. This is why utility workers can do their job on 500kV power lines that are energized.