Go up there and tell us what you see in great detail. Describe the surface, edges and penetrations. Check the actual slope of the roof and verify that it drains (it should not be "flat" and water should not pond). A photo would help. What part of Maine?

Dixfield, Maine - near Rumford/Farmington area. Oxford County.

There are drains - or at least there are pipes in the upstairs rooms that go to the roof. I haven't gone up there yet but I will take photos when I do.

I don't reckon there ought to be anything penetrating the roof, as it's a fairly "dumb" building - there's nothing special going on there. The furnace vents through a basement window (!) so there's no chimney.

You usually learn a lot when you finally go up on the roof.

Henry's wet or dry roof mastic and some fiber mesh tape but the fixes will only be temporary and you will be going on the roof often to check on the fixes. Try to find a roofer that will work with you financially ,(like a monthly payment plan), and get it fixed right. It will only lead to more costly and bigger problems down the road if you try to wing it.

"...whatever is normally installed on a flat roof."

Bitumen (AKA 'torch down'), double coverage roll, hot map & felt, EPDM, and some others I cannot remember at the moment.

The only way to tell is to go up and look at the stuff, especially and edge.

Bitumen membrane is pretty common on residential flat roofs.
The seams often fail and can be patched with butyl caulk.

Double coverage roll has 1/2 the roll width graveled and the other 1/2 plain. It is lapped and hot mopped with the gravel side up but lapped over the plain side of the previous piece (giving 2 layers of felt everywhere for the double coverage).
It can be repaired with cold patch, but the actual leak may be many inches away from the break in the roof itself.

Hot mop & felt is not that common in residential, and is easily identified by the gravel meant to protect the tar from sunlight.
It can be very hard to repair since a break can allow water to travel many feet between the layers before finding an exit.

EPDM can have the same problem with a hole in one location and the leak inside many feet away.
It can be patched easily and permanently is the defect can be located.

Thanks for the last couple of posts. I do intend to "wing it" (non-negotiable. don't waste your time. there's more to this situation than I wish to post on an online forum) but intend to be well-educated before I even start.

I haven't had time to go up on the roof just yet, as I've been busy with a few electrical wiring projects. I should be working on one of them right but I'm a bit steamed as my "assistant" decided he'd rather hang out with his friends and get drunk than crawl around in an attic. Can't say as I blame him, and I've left as well. I'm not doing that much labor for free for a "friend" who can't be bothered to help.

Since you have mentioned membrane roofs, may I ask what's the best line of attack (right way to repair) for a torch down membrane roof (very flat roof with "shallow" slope) that was topped with aluminum roof coating? A friend has some leaks on a 6 yr old addition that I can't figure out.

I patched some visible hairline cracks in the aluminum coating with plastic patch and/or plastic roof cement (tube), but it hasn't solved the problem. The most problematic is adjacent to a register that penetrates from the roof (ac installed up on top), but I'll be darned if I can see what's causing it.

Don't ever put off fixing a roof leak. The first step is to go up there and look at it. Do it now before you lose the ceiling!

The ceiling in this building is already a loss.

It's about one step up from a crackhouse. The landlord doesn't give a damn. We could probably sue him into oblivion but don't stand to gain much by doing so.

The only goal right this second is to make the place remotely inhabitable for the winter.

Any roof repair would start by looking at the roof and since you don't seem willing to do that I'm dropping out of this pointless discussion.