Phone lines aren’t working, but DSL is fine.

Usually with the module MPOEs there's a Test jack connector. You can disconnect that, which disables the interior phones, connect a phone to the Test jack and see if you can get a dial tone. If you can, then the problem is somewhere in your house wiring, which would be your responsibility to troubleshoot and fix. If you can't get a dial tone with a direct connection to the incoming voice line, then that's AT&T's problem.

However, if DSL works fine and it's just voice, then maybe there's a faulty filter or bad wiring at the Demarc. What I did when we had DSL a long time ago, is connect the DSL filter to the Test jack and then the house lines to that. I ran a dedicated line from the incoming wiring (on the Telco side) to the room where I installed the modem. That way, I had a dedicated line to the modem and the rest of the house wiring was filtered at the MPOE. Worked great for years.

If you slide the bottom large orange latch it will swing open, disconnecting your homes wiring and you'll see the jack. 

What do you mean by the phone connected to the modem works?  Did your service change?  Is this a new modem?  Is it the same number as the landline?

Thanks! I was referring to the handset plugged into the back of the modem. I’ve tested it with multiple handsets, and all work fine.

I was afraid of messing with the orange thingy because I didn’t know what it was, but I opened it up per your suggestion and found the jack. Again, I tried multiple handsets, and got no dial tone. So, I guess that answers the question of whether or not it’s external. 

If you have DSL the phone wouldn't be plugged into the modem, they would be plugged into wall jacks equipped with filter or a whole house filter that I don't see in the NID.  If your phone is feed through the modem you don't have DSL you have digital phone service.  You said in your original post that the phone lines (plural) have all stopped working,  so do you have more than one phone number in your home?  Or do you mean the phones connected to your homes wall jacks don't work? If you only have one incoming number and at one time the homes phone jacks worked than they were being feed out of the modem, which is probably why there's a splitter on the back of the modem.  I think you need to start from the beginning and tell us exactly what the trouble is and if you have multiple phone numbers and if the jacks use to work what change so now they don't.  Right now what you've stated so far is very confusing.  Also what is the beige phone cord for?

I know! It’s almost like I don’t really know that much about phones!

As I wrote up front, we have both DSL and regular, old-timey landline phone service with handsets plugged into (DSL filtered) wall jacks.

I don’t even recall why that one handset is plugged into the modem (the beige cable) but it’s not a recent development. The other phones still worked as of a month or so ago. And as I noted above, I verified that the test jack outside the house isn’t working.

I spent an hour this afternoon on the phone with AT&T, and they’re sending someone out.

It turns out that I’m King of the (Edited per community guidelines).

I used to have a separate phone line, as I described. But when they upgraded my Internet service, they switched me over to VOIP. And at some point—likely a while back when I was futzing around trying to get my printer to work—I unplugged the cord connecting the splitter on the back of the modem to the wall phone jack, thereby disconnecting the rest of the phones in the house from the dial tone.

In other words, duh.

Managed to work it out with the service guy before he even arrived.

Thanks, all!