Although I’m a big fan of multi-channel scopes and Hall-effect current probes, a dirt-cheap single-trace oscilloscope can get you quite a ways to the goal:

That’s a genuine JYETech DSO150 powered by an 18650 lithium cell and a boost converter set to 9 V. Make sure you get a genuine DSO150 from an authorized seller, rather than one of the myriad knockoffs; it doesn’t cost much more and tends to reward the right folks.
Anyhow, battery power means you can connect it directly across components to measure what would otherwise be a differential voltage:

That’s the voltage across R1, the 39 Ω LED ballast resistor in the discrete LM3909 circuit running from a 1.5 V supply. Divide the 314 mV peak by 39 Ω to get 8 mA of LED current.
The voltage across C1, the timing and boost capacitor, looks like this:

So the cap adds half a volt to the supply in order to put 2.0 V across the LED, which accounts for the relatively low current; the green LED has a forward drop of about 2.2 V at 20 mA and 1.9 V at µA-level current.
For completeness, the voltage across the LED:

So, yup, the LED really does see 2.0 V. I love it when the numbers work out.
Crank the supply to 3 V and see this across R1:

The LED current is now 1.23 V / 39 Ω = 33 mA.
The capacitor just barely enters reverse charge:

Pop quiz: what voltage to you expect to see across the LED?
I’ll leave further investigation to your imagination, but for low-frequency analog work, you can do worse than a DSO150.
Looks like a fun and useful scope to build. One’s on the way.
The bandwidth is too low for “real” digital signals, but it’s good enough for audio-frequency circuitry and suchlike. Adding the lithium battery + charger + case turns it into a very handy portable package … and I just added a USB serial board to capture its trace data dump.
For Mk I and Mk 1a power, I ordered a 9V wall wart and a 9V battery adapter.
I really need to power up the ancient HP 1MHz dual trace scope. It worked last time I tried, but I’m not sure how many years ago it was.
Is there going to be a post on the USB serial board/data dump? Asking for a friend. [grin]
The trouble with boatanchors is you must fire up their pony engine to get ’em started: all that work for 1 MHz!
And, yeah, the dump on the scope dump appears tomorrow.