A lot depends on how much power is available on the circuit, and how much your servers need. If you only have 1 phase (small computer room), you have no choice. As you get close to the rating of the circuit, expect more power supply failures and circuit breaker pops. Try to not to exceed 80% of maximum circuit amp rating. Consult several licensed, commercial electricians to design your setup if you want to do a pro job.
In a small office, your panel may have residential split phase power - 120V on most breakers, but some 208V breakers, too. This is sort of like having 2 phases - 1 phase is 180 degrees out from the other. The phase difference doesn't hurt your gear one bit.
In a bigger datacenter, you often have a PDU or power panel that has 3 phases, 120 degrees from peak to peak. When we talk of balancing phases, we mean balancing the current on each phase, not the phase angle.
Balance the loads by putting different power supplies on different phases. This avoids filling up one phase with loads too quickly. When circuits are loaded near max, balancing also helps prevent accidental breaker trips when multiple devices power up due to inrush current. It also prevents server outages if 1 phase's breaker does indeed trip due to a wiring fault or catastrophic power supply failure. Do the arithmetic - in a redundant setup, you must run each phase less than 40% of load to allow 1 phase to die, and still have some room for inrush.
If you have multiple phases, label your breakers and power outlets under the floor and your power strips in your cabinets with the phases. If you don't have metered power panels or UPSes, count the rated amps of each supply, but bear in mind that server vendors offer options on whether 1 supply is idle or if both supplies share the load. My opinion is, share the load. If you want to geek it to the max, you can use actual DC loads on each supply to calculate. This is why people buy metered UPSes and PDUs - so much easier to eyeball.
As far as totally different sources of power, the only place I've seen that is in a hospital after a hurricane Katrina, where power feeds in the neighborhood were unstable. They got separate feeds from separate commercial power substations, and used switchgear (big automatic switches - hundreds of amps @ 240 or 480V) to detect and switch over. Very expensive. Power did burp when we lost a source, but our UPSes smoothed it out. Lights would dim or go out completely, but the servers stayed up while PCs on non protected power rebooted.
It's important in such cases to have isolation transformers and well installed star grounds (all rack power grounds returning to a single ground point under the floor). Power switchover can send differential spikes down any phase and neutral, and the isolation transformers prevent the UPSes and servers from seeing most of such spike. You can still get sags and surges, though - longer rises or dips in power. Again, the UPSes protect you.
Low cost surge protectors die to protect the circuit - do not rely on them or be prepared to change them out.
Power gear vendors, like APC, server and network gear vendors have white papers that explain more of the technical details, but you need to wade through a lot of marketing hype. They often don't help smaller sites much.
Here's a paper I googled up on problems caused by unbalanced phases: