The reason you use a relay is because it allows for Bi-directional switching, same way a normal mechanical switch does because it removes the part where current is supposed to pass. Mosfet could be used to switch AC, but it's rather complex and redundant. a single one is not useful at all since it still passes current from it's source to drain for N-channel mosfets and the other way for P-channel ones. this can be seen in the way it's shown in schematics (the body diode next to it).
What you want to use is a TRIAC, widely used for switching main's rated devices (like "single" phase motor speed control applications and dimmers used to control the lightings in houses). However, your control device is not to be connected to the triac directly, you should drive it with a optocoupler (look at MOC3021 and other MOCs). make sure the continues current exceeds the normal current you would require for your devices and also include a snubber or a voltage limiting mechanism (like TVS diodes) to supress voltages of switching inductive loads.
It's worthh mentioning controlling TRIACS takes some playing around, but if it's a simple on/off application it should be fine. to reduce the current produced from disconnecting inductive loads, there are optocouplers with zero-cross detection circuitry included, which you could use. If it is a dimming situation, you need zero-cross detection. this would add extra cost to your design (one extra optocoupler). you can use the output of the zero-cross circuit as a trigger for your uC as a phase calculation for your PWM (like if 100% brightness, turn on the TRIAC immediately after getting triggered, if at 50% turn on the TRIAC after 5ms in a 50Hz system).
All in all, if you want dimming, look at TRIACs and built modules available in the market if you don't have prior experience, but if dimming is not a necessity, use your relays and supress the current from the coil using a free wheeling diode. However, as Bimpelrekkie mentioned, if the design is good you should not have a problem.
Note: Not only a small mistake in high voltage handling could kill all your electronics (high voltage connected to your low voltage side) but more importantly it's lethal and VERY VERY dangerous without prior training and experience, and I strongly suggest you avoid handling mains if you are not properly equipped with the knowledge and background