I recently installed Linux (Debian 9 i386 stable) on an old iMac: early 2006, iMac 4,1, coreduo. When it boots, it works great. But that's the rub: it really doesn't like to boot.

I get past the "Welcome to Debian!" message and it looks like systemd starts spitting out its usual messages. And then it usually locks up shortly after "Started udev Kernel Device Manager." It spits out a few more messages, usually about ACPI detecting the power button, but it isn't consistent when it locks up. For example, here is what shows on the screen right now:
I just power-cycled and got the same, except no "Raise network interfaces" message.

Sometimes I get a few more messages. And sometimes it boots up without any complaints at all. And when it does, it behaves just fine with either KDE or LXQT. Usually power-cycling after a successful boot continues the luck---it keeps booting fine, whether I reboot or do a shutdown and then turn it on again. But if I leave it off for a few hours, then I'm back to locking up.

I tried resetting the SMC and NVRAM; it booted once after resetting the SMC, but then that had no effect on later boot attempts, so I suspect that was just coincidence.

I've tried a few different kernel parameters with no consistent luck. acpi=off prevents the machine from even getting to systemd. I tried a few of the PCI-related kernel parameters, but none of them seemed to do much, and I'm really not sure which ones would be appropriate anyway.

Any thoughts? Kernel parameters to try? Magical incantations? I'd even be happy with "Try XXX; I don't know why it would matter, but try it anyway..."

Eventually I'll probably just give up. But it is just annoying me that this machine seems to work just fine (especially good enough for my kids, who don't have very high demands yet), except that I usually can't get it to start.

For what purpose do you want use the iMac?

Maybe a Desktop distro like Opensuse Leap 15.0 or Ubuntu does fit better.

I've been using Debian on servers and desktops for the last 15 years or so, so I'd prefer to stick with it.

I can't say for sure, but it really does sound like hardware failure of some description to me. Have you tested the RAM with something like memtest86+ ?

I'd try and verify that the hardware itself is OK.

Thanks for your suggestions.

Following up on jsbjsb001's suggestion that it is a hardware problem, the machine was running OS X 10.5 fine before I installed Debian, so I am a little doubtful that there is a problem with memory or hardware. However, I did run memtest86+, which reports no problems with the memory. Based on when Debian hangs, I would guess it is more likely a problem with how some part of the OS is interacting with (a very old) piece of hardware.

Following Keruskerfuerst's suggestion that a different distribution might work better, I tried to fire up Lubuntu 18.04 off of the install DVD. It chugs away for awhile with the splash screen and its five lights that alternate between blue and white. Then it stops: lights stop changing, and no disk activity. I.e., it hangs, but with rather fewer details than Debian gave me (-: So I'm not sure the distribution is the issue, though since Ubuntu is based on Debian, perhaps this wasn't the best test. But I'm getting a little reluctant to keep burning DVDs that I just have to send to the landfill (apparently this beast does not boot from USB).

Decided to try to install Ubuntu MATE on an antiquated iMac 4,1 (early '06 edition). I got through installation perfectly fine with the help of rEFInd, and the machine runs beautifully in Live mode off the USB drive, as intended.