If you could only own one pair of shoes, what would they be?

Only two pairs of shoes come to mind when looking at your description.

1) "New Balance 574": http://www.amazon.com/New-Balance-Mens-574-Sneaker/dp/B001AO21Q6/ref=pd_sbs_shoe_2

2) "Adidas Samba": http://www.amazon.com/adidas-Mens-Samba-Classic-Soccer/dp/B0007QCNGG

I'd say its fine to deviate from this a little, but these are the two pairs of shoes that have been on my feet for most of my life.

With Greg on this one, Merrel make the best non living things I've ever had the pleasure of putting any part of me in.

If you had not included the word "running," I would have said Dr. Marten's. So, I am forced to agree that if you want one to do everything, the answer is Merrell (I own their sandals and they are great). Or you can just get yourself the classic Converse Chuck Taylor All Stars.

Clarks Code. Incredibly lightweight slip-on (great at the airport), but still very durable and sturdy as well as nice looking.

http://clarks.zappos.com/n/p/dp/42710112/c/2462.html

Red Wing 1178 Roper Boots

Gotta love my black on black Vans slips.

Have you checked out Tretorn Great looking and versatile. Not to mention extremely reasonable!

I am a boot guy pretty much year round. Red Wing Gentleman Travelers are my standard footwear.

I would check out Onitsuka Tigers particularly a pair of "Ultimate 81s""http://www.asicsamerica.com/onitsukatiger/#/collection/product/HN201-4705/
They are light, comfortable, breathe well, and look great. You can get away with them at any occasion, and if you get them in black and white you could probably even pull them off at work.

anything FABI

Blundstone 500 boots, http://www.zappos.com/blundstone-bl500-stout-brown. Waterproof, comfortable as hell, can be worn with almost anything; these boots are just the best. No matter how messy you get them, just wipe them off with a wet cloth and they look "good as new". Great all-purpose shoe.

I am pretty much a Clarks and a Born kinda guy, but I would like to try some Merrell's.

Puma California: http://www.kicksonfire.com/puma/puma-california/

Nike Air Force 1: http://www.footlocker.com/catalog/productdetail/model_nbr--105443/sku--24300657&SID=8732&inceptor=1&cm_mmc=SEO-_-Feeds-_-Froogle-_-null

This is just pushing it....there is no shoe that will work for all of the above, if you weren't to include running than there are plenty of great options from Puma, Adidas and Nike among others to choose from, but running shoes are ONLY for running! they look hideous for any other occasion...not even when paired with the coolest jeans and T. On the other hand any shoe that is not specifically meant for running will not support and protect the knees the way it needs to + the running will ruin the shoes and they will look bad when wearing them for fun...
http://boaznyc.blogspot.com/2010/01/5-quick-ways-to-express-yourself.html

Adidas Sambas would definitely be my choice.

brown ones

Just about any Asics running shoe.

http://www.asicsamerica.com/products/category.aspx?PARENT_CATEGORY_ID=250001675

Sorry, better link
http://www.asicsamerica.com/products/category.aspx?PARENT_CATEGORY_ID=250001538

These would be the only boots I'll ever need if I had to choose
http://www.wolverine.com/US/1000mile/#/Rust/

They may not be the most popular fashion choice currently, but the Vibram FiveFingers are the best and most comfortable shoes I've ever owned. Been wearing them for two years now, and my KSO Treks in black don't attract quite as much attention at the pub as some of their standard fitness varieties!

http://www.vibramfivefingers.com/products/products_kso_trek_m.cfm

Adidas Samba

I have found that lo-top black Converse Chuck Taylor All-Stars rise to almost any occasion. They are not high style; they are not high performance. But once you adjust to them, they command respect with a sport jacket or below cargo shorts. I usually wear them five or more days a week (I've got some red ones to break the monotony).
Straight men will compliment you for wearing them.

the shoes I wear the most are my LL bean brown leather boaters, they look good and are comfortable, granted they are not my most expensive pair of shoes but for every day activities I think that they are the best compromise for style and comfort.