Linux is most common at the low end, where its strong built-in networking gives it a leg up over less capable embedded OSes that used to be used for tasks like this. Basically, using Linux saves companies considerable money vs. licensing an OS capable enough to match it for the purpose.
Cisco IOS is not based on Linux, or on any other common OS, that I know of. It may well be entirely custom. Cisco is certainly large enough to have accomplished that. It can't be Linux-based, since they'd have to be giving away the GPL'd parts at least, and they aren't doing that.
Cisco's largest competitor in the router arena, Juniper Networks, uses Junos on most of their equipment. It is based on FreeBSD.
As for your Belkin router, the F5D8235-4, it is indeed based on Linux. That link takes you to a page linking to tarballs containing the GPL source code each of their products uses, as is required under the GPL. I had a peek inside the F5D8235-4 tarball and found a copy of the Linux kernel sources inside.
To answer this question for other routers, you can do what I did: look for a place to download GPL'd sources for the router, or look for copies of the GPL license in the product manuals or downloads. The GPL also requires that they tell recipients of products containing GPL-licensed software about the license. The fact that you got a copy of the GPL doesn't mean you have a product based on Linux, but it does mean you should be able to download the source code from the company somehow, and thus answer the question by examination.
Another method is to try telnetting to the device. Many Linux-based home gateway routers will respond to Telnet. If it lets you in, some elementary exploration can answer the question; a
uname -a command, for instance.
Another big clue is discovering that it runs BusyBox. BusyBox runs on other OSes besides Linux, but it's very commonly used on embedded Linuxes, whereas the more minimal stock BSD toolset means embedded BSDs tend to go with the standard, rather than the even more stripped-down BusyBox tools.