There are actually 3 Nettos that I know if in Leeds, the Seacroft one as somebody already mentioned, also there's one on Kirkstall Road - just 1 bus stop past the Warner Village (well it used to be called the Warner Village anyway....dunno if it's still called that....where Hollywood Bowl is and Vue cinema). That is the biggest Netto - I go there regularly.
and the third Netto is in Meanwood, just across the road from Aldi.
Other than that, as people have already said, there's Aldi in Meanwood (see above), and Lidl in Gipton. I go to both but then again I have a car.
Other supermarkets that I know of are:
ASDA in Headingley
ASDA on York Road next to Killingbeck Cemetary
ASDA in Pudsey
Tesco on Roundhay Road
Tesco EXTRA kinda next to York Road type thing (bit hard to explain)
there's a very small TESCO in Crossgates shopping centre
Morrisons in town - Merrion Centre
Morrisons on Kirkstall Road
Sainsburys on King Lane, Moortown
Safeway in Oakwood
Kwik Save in Harehills
and I find it very highly unlikely that you will be shopping here, but there's a Marks & Spencers in town, and there's a smaller one in Moortown (Moortown corner parade, next to The Bull pub).
I'm sure there's more supermarkets in Leeds but those are the only ones that are coming to me off the top of my head.
Oh yeh, just remembered there's a supermarket in Chapel Allerton aswell but I can't remember which one it is....it USED to be Kwik Save but it got changed many years ago. Think it's Somerfield but don't quote me on it.
Also there's an Iceland on Kirkstall Road next to Netto, and another one in Harehills next to Kwik Save.