The other answers here are giving mixed answers, because an SMPS has a complex input circuit.
During startup, there is an inrush transient. This is capacitive in the sense that it's due to the main energy storage / filter capacitor, but it will stretch over several cycles (while the inrush limiter device -- usually a NTC resistor -- is active). So, on a cycle-to-cycle basis, the waveform won't have much phase shift during this event, and it's more that the envelope of current flow is initially large, then decreasing.
Inrush also takes place somewhat later than the instant of turn-on. In the first nano to microseconds after contact closure, current flows into the mains wiring: this can occur very quickly, as the nearby wire segments get discharged from anything up to mains peak voltage, down to ~zero difference across the contacts. This occurs very quickly, so expect transient emissions up to about this many volts, and with fast risetime (similar to any other EFT transient). The current associated with this event corresponds to transmission line impedance: in the ballpark of 50 or 100 ohms. So, it's low energy, but certainly enough you can pick it up on a 'scope.
Over a somewhat longer time scale, current flows into the SMPS input filter elements, usually a CLC ("pi") filter. How much, depends on line impedance or inductance, and filter component values. For a typical supply with say 0.47µF at the input, and for a mains network similar to a LISN (50R || 50uH), this will give some ringing around 30kHz, decaying over a handful of cycles. There's also a common-mode choke which has some leakage inductance, and another capacitor behind that (and usually another one after the rectifier too), which will put some finer ringing on top the first waveform.
Conversely, at turn-off, when the contact opens while current is flowing (particularly near peak), the energy stored in all these inductances, small though it is -- gets discharged across the contacts. Typically a small spark develops, generating some hundreds of volts with a risetime in the nanosecond domain. This can spark multiple times: the voltage rings down after the transient, giving time for the contacts to open a little further, until the voltage rebounds to breakdown (at the now slightly higher voltage due to the contacts having opened further), and so on until the energy has dissipated. This describes the phenomenon of EFT (electrical fast transients), but mostly requires much higher inductances to occur -- coils and motors in the 10s of mH. So there should just be one or two sparks in this case.
In summary: the phase angle depends on frequency and operating condition. Operating condition ranges from inrush, to normal operation, to turn-off (with operation being a range, since the rectifier and PFC stage impedances vary with current). Frequency is driven by the LC components at the front end of the supply, as well as the wiring up to it.
For a relay on an SMPS, inside a machine, especially if the relay is supplied from an inlet mains filter -- it's only a concern if the few hundred volts peak of emission may upset other things inside the machine. Which, mind -- things should be well enough wired that this doesn't happen -- but if your wiring/layout isn't great, well, that's not great to begin with, but it takes two to malfunction, and you might manage to solve the problem by quieting the contacts instead of hardening the other one.
Without an inlet filter, just switching plain old mains, at whatever wiring length it may have -- not likely to be much worse. Again, there can always be something, but it's unlikely to upset anything meeting basic levels.