So the tires I have on the bike now are 700x30, but the only things I've been able to find are 700x35 or 40. Would these tires still go on my rims, or would I have to purchase a new rim as well?

Thanks to anyone that can lend some help. This winter has been rough so far.

At first guess, I would think you would be OK. Certainly a 35mm tire will fit in place of a 30 mm tire on the same rim. Whether it is ideal is another question!

Sheldon Brown (Tire Sizing Systems) suggests that a tire width 1.45 to 2x the inner rim width is good. I run 35mm tires (cyclocross and studded winter tires) on 19mm rims with no issue. Unless your rims are less than 19mm you should be good.

The next question is whether you have sufficient frame/fender/brake clearance for the larger tire (and any ice buildup). Only examining your bike will tell for this.

Hope this helps

I think they would work fine. I've exceeded the tire size spec on rims often, and haven't had any bad results. Just make sure as you're pumping the tire up for the first time that the bead is well seated or else you'll explode your tube. You would also want to check whether or not 35+mm tires would fit in your frame.

However, Schwalbe does make a 30mm studded tire though: Schwalbe Winter 118 Spikes 28 x 1,20" Wire | Ice Spiker