Dedicated power

I would go with dedicated 20 amp circuits first.

a forceful foot pedal hit on a bass drum.

Exactly what is shown here http://ielogical.com/assets/CblSnkOil/Signal_vs_ACLine.png
The drum transient occurs right in the middle of the negative A/C signal and the wire is supplying ZERO current to the +ve PSU filter bank and will continue to supply zero current until the line voltage rises above the filter bank voltage X the power transformer ratio. At that time a large current spike will charge the caps. However that spike is well below the line capacity.

Examine the "White Rabbit" track. The signal oscillates either side of 0v through the entire A/C line wave cycle. The filter caps are only charged when it is their turn on their half of the A/C cycle.

Rock music has about a 20db dynamic range or 100:1 power ratio. So a 500W amp is putting out a fraction of its total capability. The ONLY time anywhere the line ampacity is required is when driving a load resistor.

 Voltage, length of the wiring X 2 and the gauge of the wire used.

I covered that in Headroom Loss for 1600w on 14ga/120v | Audiogon Discussion Forum

Upping the gauge from 14 to 10 results in a 170mΩ decrease in wire resistance over 54 feet. The voltage ratio is  20 * log ( 0.987 / 0.968 ) =  0.169db relative to the unloaded line voltage!  see  ieLogical CableSnakeOil

It's my contention that improvements heard after a 10ga rewire are largely due to direct clean connections and not the wire gauge. If rewiring for audio, go direct and only use screw terminals on quality outlets.

My media room has a dozen outlets which were daisy chained with push terminal quick connections. This circuit is used for 2x 600w electric heaters. I'd estimate the length at about 80 feet. I had a 5-6v drop from the first outlet to the last when loaded by both heaters. Removing the quick connects, cleaning the wire and using the screw terminals resulted in under a volt drop when loaded.

If one has to choose between getting a dedicated line or a good power conditioner, which one provides the biggest bang for the buck.

First determine if you have a problem. The best answer maybe neither.

It's not just clean connections, I've had lights blink on a 15 amp circuit with powerful bass, but when I plug an electric heater into that same circuit the lights are fine no change. 

If lights are 'blinking' they are on the same circuit. Sockets and lamps should be wired on separate circuits. Likely there are other high current loads as well.

I've been doing this a very long time. Including professional recording studios. 12ga wire and 20A breakers for walls of amplifiers. Main monitors typically 1 x 20A circuit for 4 x 500 watts [250wpc]. Playback on STUN didn't dim the lights [on a separate circuit] or jiggle a peak responding line voltage monitor.

On a 14ga 15A circuit with 10 foot run to the socket, I currently have tri-amped speakers - total 315WPC - plus 2 subs 300W each. Speakers are ≈84dB/w/m and peak levels are +100dB. The REAL cannon in Telarc's 1812 barely jiggle the line voltage when measured with a DSO. Amplifier outputs scale to input levels, i.e. NO compression.