I have Canon EOS 700D and want to buy a wide-angle lens. So far I was really impressed by the 10mm ones, however the ones with at least decent aperture comes without Image Stabilization. I also looked to the 15mm ones, with IS. Ultra wide angle is a really desirable thing for me but I would also like to protect my pictures from blurring since I am not a fan of using a tripod. Which one of these lenses are better for me?
First, some terminology. On your 700D (or any of the 1.6x crop APS-C bodies), a 15mm and up is still just "wide angle". It's below 15mm that lenses become ultrawide. So if you want ultrawide, you need lenses that are around the 10-15mm focal length range. Wide angle on a 1.6x crop camera, typically means something in the 15-24mm range. Normal is around 28-35. And telephoto begins around 40mm. (For film or full frame, ultrawide would be below 24mm, wide from 24 to 35mm, normal around 50, and telephoto starts in the 50-85 range).
Whether you need a wide or an ultrawide is a matter of personal taste, but essentially going very wide means more distortion as well as more coverage, so you're basically choosing how funky and extreme you want to get with the wide angle. Also, be aware that most of these wide angle choices on Canon are only for crop bodies (Canon EF-S, Sigma DC, Tamron DiII, Tokina DX, etc.), and probably will have to be replaced if you ever plan to make the full-frame move.
Whether you need stabilization or a large max. aperture depends on how you plan to use the lens. Most folks do not find IS to be critical with ultrawide lenses, because their main use will be for landscape/cityscape shooting with a tripod. And a tripod will beat IS any day when it comes to long exposures. While it is more inconvenient, it also is a much better tool than IS for long exposures, and generally doesn't have an upper limit, shutter-speed-wise, like IS does. And a tripod or monopod (a good equivalent to IS) works with any lens you have.
In addition, good handholding technique and choosing appropriate exposure settings can take care of camera shake issues as much as IS, and IS, while convenient, is still no guarantee of eliminating motion blur from camera shake.
The only use where you might want stabilization and a large max. aperture is if you plan to shoot environmental portraiture handheld, or architectural interiors without lighting.
That said, there is one lens that it seems you have not considered which is both 10mm and has image stabilization, while being lower-cost than either of the two lenses you're looking at: the EF-S 10-18 f/3-5.6 IS STM.
Also, if you find that you are an ultrawide junkie and you'd like to go even wider, there is one class of lenses you might want to consider if the distortion doesn't scare you off, and that would be fisheye lenses. Fisheyes give up attempting to project the image in rectilinear fashion (i.e., keeping straight lines straight), and allows lines to curve (maps equisolid or stereographic). I shot for many years without an ultrawide because I had a fisheye in the bag.
So far I was really impressed by the 10mm ones, however the ones with at least decent aperture comes without Image Stabilization. I also looked to the 15mm ones, with IS. Ultra wide angle is a really desirable thing for me but I would also like to protect my pictures from blurring since I am not a fan of using a tripod.
A longstanding rule of thumb tells us that to avoid camera shake during handheld shooting, the minimum shutter speed should the reciprocal of the focal length. For example, if you're using a 50mm lens you probably don't want to shoot faster than 1/50s; if you're using a 200mm lens, 1/200s is about the limit. There's plenty of room for argument there -- some people have steadier hands and better technique than others, etc. Nevertheless, thinking about this rule lets you see the general relationship: camera shake affects wide lenses less than it does long lenses.
It's easy to see why this is if you play with the numbers a bit. On your APS-C camera, a 10mm lens has a horizontal field of view of about 97°, so a small horizontal rotation like 1° results in the image moving 1/97 of the frame width, or about 1%. At 85mm, on the other hand, the horizontal field of view is around 15°, and the same 1° rotation results in the image moving 1/15 or about 6.6% of the frame width. Now, I expect that the actual magnitude of camera shake at normal shutter speeds is somewhat less than 1°, but you can see from the example that the effect of shake at 85mm is more than 6x that at 10mm.
Put another way, you really don't need IS on a lens like the EF-S 10-22mm, but you might need it on the long end of the EF-S 15-85mm.
I can't tell you which lens is best for you, but if a wider lens is important to you, don't let the lack of IS deter you from going for the EF-S 10-22mm.