I'd like to see a bit more data, like the second channel trace, before you start taking it apart. If you take the case off that might give you a longer test time before it starts glitching. Also, turn down the intensity! The trace should not "bloom" under normal conditions, it should just make a nice readable line without illuminating more of the phosphor than needed for visibility. Don't worry at all about not seeing the rising and falling edges from the Calibrator, that is normal at this horizontal resolution and indicates that the calibrator circuit is giving you nice fast rise and fall times, a good thing.

Instead of using the scope's calibrator as a signal source, you could also try connecting the GenRad oscillator to get a sine waveform instead of the square wave. For low frequency testing you don't really need to worry about fancy connectors or precise impedance matching, just use clipleads to connect the GenRad's ground to the scope's BNC grounds and another cliplead to connect the center pin of the GenRad connectors to the center pin of the scope's BNC channel inputs. Set the GenRad to the same frequency as your calibrator output. The sine wave will not have "invisible" portions like the square wave from the calibrator does. Start with the output of the GenRad set to the lowest possible value, then gradually turn it up until you have an acceptable trace amplitude on the scope.

When you do open it up and start prowling around in there, I would suggest that the very first thing you do is to check the power supply voltages, using a DMM and the information in the SM. You have a handful of low-voltage DC supplies to check, starting with the +55V supply.  Since you have a good beam and at least acceptable traces until it starts glitching, I wouldn't even worry about the HV supply at this point. But off-voltage LV supplies can cause all kinds of weird problems, and since these use old and maybe flaky capacitors, and the problem is heat- or at least on-time related, that's where to start looking.

The LV power supply test points and voltage ranges are on section 6-8 in the SM. You'll have to look at the board drawings to find the locations of the numbered test points. Don't worry about testing the HV supply. And don't try adjusting anything yet! Just let me know what the voltages are at the test points. Be sure to set the scope's controls as required for the PS testing. The voltages are referenced to chassis ground, so use a nice sturdy cliplead to "permanently" connect the DMM's Minus input to the chassis. Then carefully probe the TPs with the DMM's pointy red lead. Be very careful because obviously you don't want to short anything out or hit a HV point by mistake. But this is actually really an easy testing to do once you have located the correct testpoints on the boards.

that is an effect of the camera.  the brightness was only up just high enough to where the vertical traces are slightly visible.  the illumination was also down all the way.  I don't know why the camera does that.  maybe I have it on low light setting or something.

The traces were blooming heavily.  If you don't turn it down, you are going to burn the screen.

Oh ye of little faith.

Well, I am going to repeat my cautionary note again.  I get the impression that Brian has no electronics experience at all and finding bad components and changing them inside an oscilloscope is not a trivial matter even if you have the service manual.

As an example:  Brian, if you switched off the power to the scope but left it plugged into the wall socket, did you know that if you touched a component with a regular soldering iron things could "blow up?"  If you didn't know that then that shows you should be careful.

Here is a way of looking at it:  From what I can see from your clips the problem is annoying but tolerable.  The scope is still working and showing you waveforms.  The suggestion is to just use it like that for now.  In six months if you truly are interested in this stuff and are learning the whole time, then perhaps try to fix it.  However, there is a chance that in six months you will have moved on and dropped the whole thing.  Then fixing the scope becomes a moot point because you will not be using it.

MileHigh

Yes, I know very well the proper way to use and work on electronics.  I'm not a complete moron.   I can solder perfectly well, know how to desolder.
I'm not trying to be rude here, but I'm not some average dummy.   I may not know anything about oscilloscopes, and I may be a beginner in learning about how more complex circuits function, but I've known how to solder and how to fix things for a long time now.   
A little background on me:  When I was a little kid, my favorite thing to do was take things apart, learn how they work, and then put them back together.  I never cared about action figures or pointless stuff like that.  I started doing construction at 13 and excelled at it, started fixing cars at 15, and excelled at that.
I've wired up multiple houses, mapped out the circuits in houses built in the early 1900's that had knob and tube wiring, rewired houses, fixed circuits, etc.
It was only recently that I got back into smaller electronics though.   I'm a little older now and don't learn as quick as I used to, and it's not as easy for me to figure things out like I used to, but I still can. 
So what I am saying here is I'm not your average numbnuts and I don't like to be treated as such.

Brian:

All is well and fine.  The soldering iron issue happened with someone well known around here about a year ago and he is a guy that also loves to take stuff apart.  My language may not always be the most politically correct but I was not trying to prejudge you.  You seem to be making progress so I hope that you fix it and then you are off to the races.  Just be careful of the high voltage.  TK is the expert there and might have some sound advice for you about that.

MileHigh

Especially DC HV, since at least AC will throw you away from it, instead of getting stuck to it. 

Brian

I am no expert but this is not correct from what I was taught while working on 3 phase 240 volts machinery.  I was taught that you never grab anything...you use the back of your hand so in case you get zapped, and your hand closes involuntarily, you are not clinging to a hv source.  Now, maybe this is only true for 3 phase circuits but...I don't think so.  This saved my life once when working on some equipment (before lock-outs were invented) with the breaker off and, a note saying to leave it off.  Well, someone was trying to figure out why their machine was not working and...yes...turned it on.  I did get bit but, I was careful not to be in a position to have my hand close around the wire.  I was also taught to use one hand only, that way the charge does not go across your heart.

Off-topic I suppose.  Sorry.

Don't worry...your scope was made with quality components and meant to last.  It could easily be as simple as one of those caps.  You will get it figured out and then you will feel very proud for doing so.  I know that I would.

Bill

I am no expert but this is not correct from what I was taught while working on 3 phase 240 volts machinery.  I was taught that you never grab anything...you use the back of your hand so in case you get zapped, and your hand closes involuntarily, you are not clinging to a hv source.  Now, maybe this is only true for 3 phase circuits but...I don't think so.  This saved my life once when working on some equipment (before lock-outs were invented) with the breaker off and, a note saying to leave it off.  Well, someone was trying to figure out why their machine was not working and...yes...turned it on.  I did get bit but, I was careful not to be in a position to have my hand close around the wire.  I was also taught to use one hand only, that way the charge does not go across your heart.

Off-topic I suppose.  Sorry.

Don't worry...your scope was made with quality components and meant to last.  It could easily be as simple as one of those caps.  You will get it figured out and then you will feel very proud for doing so.  I know that I would.

Bill

I believe we are both correct.

If the AC source you are handling is something like a wire, your hand would close around it from the reflex of getting shocked, which would in a way hold you to it, but if you get zapped by AC say, on the tip of your finger, it will blow you back, and if it's strong enough, possibly blow off your finger and a toe or two if that's where it decided to leave your body at.  I know someone that lost two fingers and two toes from HV AC.    I could be wrong here, but I believe we are both correct.

Brian

Interestingly enough, my manual, which is for my exact model, and in the correct serial number range, says that the cap on the PSU +5V rail is an electrolytic 33uf 10V..
My manual is the original one that came with the scope when the power company purchased it originally, and has the fold-out schematics and PCB layouts.  It doesn't sound like we are both working with the same copies.  Mine doesn't have page numbers in the diagrams section, either. just side tabs.

Actually the parts list in the manual I have identifies all the tantalums and other electrolytics just as "electrolytic". The page numbers I'm citing are the pages of the .pdf file that I downloaded from the electrotanya link I provided earlier. It's a real pain that the manual itself doesn't have page numbers. Perhaps the easiest thing would be for you to download that manual too so we can refer to page numbers-- I obviously don't have the paper copy like you do. You have the advantage of seeing the good drawings of the boards, the ones in the .pdf file kind of suck. But it is nice to be able to refer to page numbers--- you could look in the .pdf file at the page numbers I cite, and then find the correct page in your paper copy.
The box referring to the tantalum decoupling caps is in the Power Supply troubleshooting flowchart for the low voltage (15, -8, 5 v) supplies. The schematic is the "power supply" tab , marked on the tab #12 in a diamond box. And the box referring to the tantalum decoupling caps is on the tab marked "+15V, +5V and -8V supplies Troubleshooting"

Yes I have a 30W fine tip, and also a 6-8W fine tip battery powered one.  While desoldering, I tried the 6-8W and it doesn't wanna stay hot enough so I've been using the 30W.  I don't have a desoldering vacuum tool, so I made a solder sucker.  On the ones where I have access to both sides, I've just been heating one side and blowing the solder thru from the other. been working great so far, but I'll have to suck it out for these caps cuz I'm not taking the boards off again.  I already took the trigger and sweep boards off and cleaned everything up and inspected them today so far.

I may actually have replacements for these on hand. If I do, should I just go ahead and swap them out, or at least the PSU one first? Just in case it is causing a "ripple" or overvoltage, so it doesn't cause any further damage? Or is it unlikely that it would damage anything else running it with the possibly bad cap?
I don't have any page numbers like that, only section tabs and then the section number - page format. What section and whats on the page beside it?

Good on the iron. I'd check the voltage at the testpoint first, (read the box on the troubleshooting chart about how they behave as the scope warms up), then I'd replace the caps if you have the right capacitance values. You can use higher voltage ratings of course, if there is room physically. You may even be able to use other capacitance values as long as they are close: the originals are 20 percent tolerance, really cheap tantalums.

Hopefully I have some in my stash as well.  I have quite a large variety of caps of all kinds. For something that inexpensive, I will order from DigiKey since we know they have high quality components, but for future orders or more expensive things, take a look at OnlineComponents.com and let me know what you think.  When I ordered MJL21194's I found they had the best price on em, and they were actually the 21194's and not the G's, and are genuine ON transistors. Let me know what you think when you get around to checking them out.

But for the meantime, everything is back together, the cover is still off.... I'm going to check my stash for replacements and wait for your reply......

I'd say, let the scope cool completely, then power it up, test the  +5V TP1558 with the DMM right away, then let it warm up to the point where the glitching gets bad and test the TP again to see if there is any difference. Then, regardless of what you see, shut down and replace the caps if you've got suitable replacements. Then repeat the process starting with a cold scope again.

Another thing you might like to try, since the transistors are mostly all in sockets, is to wiggle them or tap on them with a wooden or plastic tapper and see if the display responds in any way. If you find one that does cause a change in the display when you tap on it, shut down the scope, carefully pull the transistor and re-seat it, maybe the it wasn't making a good connection and reseating will fix that sometimes. Lather rinse repeat.