Arduino relay buzzing sound at higher voltages
For reference and to protect against edits, here is the schematic being discussed:



The major problem here is that there is not enough input voltage to the 7805.

The 12 V secondary is center-tapped, so puts out 6 V RMS on each end relative to the center.  That means the peaks are 8.5 V.  There is one diode drop in series, so the input of the 7805 sees peaks of about 7.8 V.

7.8 V is enough for the 7805 to produce regulated 5 V out, but just barely.  Since that's at the peaks only, the voltage out of the diodes will be lower most of the rest of the time.

What is happening is that as you slowly crank up the voltage, the peaks into the 7805 get just high enough for it to produce just enough voltage to trip the relays at those peaks, but not at other times.  That's why the relays buzz.  They are being driven at twice the power line frequency, because there are two peaks per cycle.  Turn the voltage up all the way.

However, even at full voltage you still have a problem.  22 &micro;F to hold up the voltage between peaks is a joke.  1 mF on the output also doesn't make much sense since that only helps after the regulated voltage has already started sagging.

Flip the two caps around, assuming the 1 mF cap can handle the input voltage.  It should be rated for 10 V at least.  It would also help to change the diodes to Schottkys.  That would give you maybe another 300 mV, which you desperately need.

Let's see what kind of load this power supply can sustain with a 1 mF cap on the unregulated voltage and with Schottky diodes.  Let's say the peaks go as high as 8.1 V out of the diodes now.  You didn't say what your line frequency is, so I'll use 50 Hz in this example.  For simplicity, we'll say that the cap gets charged instantly every 10 ms.  Let's say the 7805 needs 7.5 V in to keep the output regulated.  That means the voltage on the cap can be allowed to drop 600 mV between peaks.  The maximum output current to stay in regulation is therefore (600 mV)(1 mF)/(10 ms) = 60 mA.

If all the relays you ever want to power at the same time require more than 60 mA total, then you need to re-think this power supply.  And, note that this is after the changes already described.