Try loading the supply with something.
An unloaded supply could have unexpected hickups.
Try a ~100uH toroid inductor in the positive line, with behind it some capacitance to ground.
Got one in my junk box here from a vacuum cleaner triac speed control board.
Leo..
This looks to me like its simply radiated pickup, its common mode noise and your chasing ghosts
Try shorting your probe tip to the ground clip and you will probably see that the noise is still there even though you are theoretically measuring zero
The probe lead acts like a receiving antenna, all the capacitors in the world wont do jack shit and theres not a great deal you can do except shield
Watch this
Like I say short out the probe and see if the noise is still there
Ah, looks like I've fallen into a "trap for young players" (I say this as I tap-tap, tap-tap my oscilloscope screen in classic dave jones fashion).
I didn't know about "common mode noise" but it seems like that's what I have. In that video by Dave Jones, his scope display looked exactly like mine, showing regular spikes from the power supply. Moving the probe wire around changes the noise level, so it looks like it all noise. I don't know what is causing the noise I'm picking up, everything in my room is turned off or unplugged, but I'm in a dorm style residence, so there may be something on the same line in another room.
Here's a screenshot of my scope, connected to the power supply, with the power supply unplugged sitting on my bench.
It's not much noise, but it's consistent and sufficient to trigger from.
Thanks all for your suggestions and help. I really appreciate your effort in helping me sort this out. I'll post a picture of my power supply when it's finally done.
John_S:
Ah, looks like I've fallen into a "trap for young players" (I say this as I tap-tap, tap-tap my oscilloscope screen in classic dave jones fashion).I didn't know about "common mode noise" but it seems like that's what I have. In that video by Dave Jones, his scope display looked exactly like mine, showing regular spikes from the power supply. Moving the probe wire around changes the noise level, so it looks like it all noise. I don't know what is causing the noise I'm picking up, everything in my room is turned off or unplugged, but I'm in a dorm style residence, so there may be something on the same line in another room.
Here's a screenshot of my scope, connected to the power supply, with the power supply unplugged sitting on my bench.
It's not much noise, but it's consistent and sufficient to trigger from.
Thanks all for your suggestions and help. I really appreciate your effort in helping me sort this out. I'll post a picture of my power supply when it's finally done.
See Post #8.
Tom.. 
HI,
I would still keep my eye on the noise. Your P-P voltage was ~ 800mV where in the video the P-P was 19 mv.
Common mode noise is noise that is common to both leads of the power supply (both the input and output).
Consider a case where you have a power supply with a perfect output of 12V. By perfect I mean the output is 12V with no noise. Now let's assume both the 12V lines (i.e + and -) are going up and down with respect to something else (for simplicity we'll assume the something else is ground). This noise that is common to both + and - is called "common mode noise"
I hope my description wasn't too confusing.
BTW the USB core I suggested you try is for common noise in the USB device.
John
I have my power supply functioning. I want to do some upgrades as I have time:
-a linear regulator for the final stage rather than a switching regulator,
-upgrade the outputs to 30V/3A rather than 12V (install a boost converter before the final regulator),
-and use mosfets to switch the outputs rather than relays (they're bouncy...).
Here's picture of the front panel:
It has 2x floating outputs, which can be tied together (the "tie" button) for increased voltage, or +/- supplies, depending on which one is grounded.
2x USB chargers, with voltage, current, and power display.
4.096V and 1.25V references, and 1000, 0.1% ohm resistor for multimeter checks.
Continuity tester.
Vacuum pen attachment (for smd pick and place).
On top, a 120V outlet, and a 12V cigarette outlet.
Back view:
"DC in" powers the left output. Can be used with a battery pack for remote operations.
Inside view:
12V, 10A DC supply burried under everything, powers the left side, USB output, and relays.
12V, 5A DC supply on right side, powers right side and fan.
3x switching voltage regulators (which I want to replace with linear ones).
Burried under the wires on the left side are 2x Atmega328's, one runs the buttons, relays, and leds on the front panel, and the other one controls the USB display.