Not sure what exactly you are asking but it would seem the answer depends a lot on what your modem is connected to.
Most cable infrastructures contain powered elements throughout the neighborhoods - if one of those nodes that feed you loses power then your cable goes down. FIOS is completely passive so as long as the Verizon central office has power (or is running on battery/generator) and you provide power to the FIOS device (ONT) on your end then yes you will have internet connectivity...I tested this theory just a few days ago.
If you are using DSL or an analog/dial-up modem then you should be fine as long as your telephone company has power and you keep the device on your end powered by an UPS.
If your question was more along the lines of what devices on your end need to be plugged into the UPS to keep your trading station up and functioning during a power outage, I guess the list would be: PC, monitor, switch (if applicable), router, modem or equivalent. As someone else said, the best way to test that theory is trip the circuit breaker - good idea to do that from time to time (during off hours) anyway to make sure your UPS can handle the load.
If you are going to do the power fail test I recommend leaving the PC at the BIOS setup screen during the test - it draws basically the same amt of power as when its running the OS and in case the test fails (PC shuts down) it won't be in the middle of writing to the hard drive which can cause a lot of problems.