My favorite EDC bags:
An Eagle Creek Flight Bag has been my EDC for over a year. Discontinued, but there are lots of similar bags. It is your basic brick-shaped flight duffle with a D-panel opening, pockets front and rear, and a pocket on each end. Those end pockets are the best feature-- my insulated coffee cup goes in one and a stainless steel water bottle in the other. The current Eagle Creek Tarmac Flight Bag is near identical except those end pockets are zippered pockets rather than an open sleeve. I do like Eagle Creek for the quality. Travel Pro is another favorite of mine.
BTW, those stainless steel water bottles make for a great sap if you on the bus or subway and the SHTF. Imagine getting a kilo of liquid-backed stainless steel in the noggin-- you're going down hard
Timbuk2 messenger bags. A medium is big enough. My misgivings with most messenger bags is the lack of a water bottle pocket and the top is just a simple flap. If it rolls over, stuff falls out all over. I do use smaller zippered sacks inside any of my EDC bags to help organize and help prevent loosing small items. The Timbuk2 bags do have some pockets inside. The ones made for laptops are a whole other deal and have better pockets all-round. If you aren't carrying a laptop, you can stash other breakables/scratchables in the sleeve. A large frame auto would snooze soundly in there

You can haul a house in the larger ones-- too much to be comfortable really. I've been a photographer for 30+ years and my rule is: "what ever size bag you buy will be immediatly filled to capacity, so don't buy one larger than you want to haul full."
The North Face Base Camp bags are made of PVC tarpaulin. The small one is about briefcase size and is really a man-purse. It doesn't have a water bottle pocket, but I hung a web-strap water bottle holder that I got at REI on the shoulder strap and that is a pretty good compromise. I use it for a camera bag, but it would make a great EDC if I didn't carry some much stuff. The larger one is more messenger bag sized. All are really tough and the tarp material doesn't get as dirty.
Computer cases make good EDC bags, which is just what they were made for. You can find excellent ones by Dell and Targus in thrift stores for $5-$10. The only down side is they are a computer case and theft prone. Many of the Targus bags are lifetime warranty. I bought a leather one in a thrift store that was missing the shoulder strap. I called Targus to buy a replacement and they sent one FREE. Brenthaven is another computer case maker that has good bags for EDC. They put their logo on the inside, which I really like.
Filson makes some great traditional style field bags. Tumi makes excellent ballstic nlyon briefcases (and they cost like blazes). The Land's End canvas briefcases are an excellent buy and will take all kinds of abuse. I bought a used one in a thrift store and hauled tools in it for years. They make ballistic nylon ones too. I have a couple in my luggage storage (all came form thrift store and yard sales). Domke is a maker of tough pro-photographer bags and they offer a few briefcase style man-purse bags, with prices around $100. Tamrac and LowePro are other camera bag makers that have turned out good EDC bags too. There's no reason you can't use a camera bag for EDC. A camera bag is a tool box for a lot a of people.
With any EDC bag, I have passed on military style bags simply to blend in. As other mentioned, the CountyComm bags are a great buy. If you can find a real Israeli Paratrooper bag, they make a good man-purse. I found one imported by Banana Republic and swapped out the shoulder strap. It makes a great camera bag and will haul your lunch and commuter toys. I can't stand the cheap Chinese copies. If the insignia is printed rather than embroidered, you know it's a knock-off.