Linear vs. Switching Regulators
There is no general truth, switched regulators aren't always the best choice.

Using a relay by itself has nothing to do with the need for a switched regulator. What is the case is that a relay uses some current when activated. Depending on the relay this increased supply current can be a good reason to use a switched regulator.

For circuits with a low supply current, for example less than 10 mA, using a switched regulator makes little sense. The power loss due to a linear regulator will be small at 10 mA or less.

For large currents, for example 100 mA and higher, it starts to make sense to use a switched regulator as the power dissipation of a linear regulator is becoming significant and you might need a heatsink. This might add more cost than using a switched regulator.

Of course it also depends on the difference between regulator input and output voltage difference. When the difference is small, a linear regulator might be a better choice.

Switched regulators produce ripple at the output voltage this is due to their switching nature. I would not call this "noise", a better word is "switching noise". Noise is inherently random and this ripple is not.

Most circuits especially digital/logic/micro controllers are quite immune to this ripple so there is little need to worry about it. Usually only analog circuits like RF circuits, filters, ADCs are sensitive to this ripple so then a linear regulator or additional supply filtering might be needed. It is also possible to us a switched regulator to for example, drop from 12 V to 5 V and then use an LDO (linear regulator) to drop 5 V to 3.3 V for a sensitive circuit.

If the a signal (not noise!) from the relay resets the micro you have a design issue! You should use a flyback diode across the relay's coil and add a decoupling capacitor near the relay and near the micro. This has nothing to do with linear vs switched regulators but everything to do with proper design.