When a phone line rings, there's 90 V RMS AC at 20 HZ on the line. It's enough to give you a jolt you won't soon forget. Thus, it's hard to miss! My favorite detection scheme looks like this. This is off the top of my head so you may have to twiddle component values a bit. Also, this is for driving logic circuits. I'll treat your specific problem in a bit.
Mostly, there is only DC or small signal AC (audio) on the phone line. C1 blocks the DC, and the R3-R2 voltage divider prevents the low level AC from having any effect.
When the ring signal comes along (90V RMS) enough voltage is developed at the juncture of R3 and R2 that some charge is pumped into C2 via CR3. Protective diodes CR1 and CR2 ensure that the output of this circuit will not grossly exceed the power supply levels and thus damage the logic circuits it may drive.
C2 and R1 have a time constant of 1 second, so one second after each ring ends, the output will fall to a logic zero again. This circuit could easily drive a counter, to count rings. A timer with a longer period could be used to reset the counter if no rings have come in within say 10 seconds.
I haven't built this, but here's my theory: C1 blocks DC, R1 limits the current that the ring voltage could cause. The ring voltage is rectified by CR2, filtered by C2, and limited in amplitude by zener CR1. Then the charge stored in C2 is slowly used to light LED1. As long as C2 is large enough (I'd start with 10 uF and experiment from there) to keep the LED on between rings, and small enough that the LED goes off within a reasonable amount of time after the last ring, you're set.
For the specific problem in the post I'm answering, (` LED should stay ON while the phone rings`) you would want to change the design somewhat. Here's how I would make a ring indicator light that stays on until the phone acutally stops ringing:
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