While most of the important communications involving ICBM operations takes place over secure hardlines and encrypted satellite communications, there are a few things your average civilian can listen to at each of the active ICBM bases in the U.S., using standard Police Scanner radios such as those carried by Radio Shack.
The most commonly heard communications are those used by dispatching trips traveling in the field: maintenance crews, security teams, launch crews, supply drivers. Each base has a radio network of repeaters set up for this purpose, needed because the silos are spread out over such a large area that the radios often do not have enough range by themselves. Usually there is a single primary channel used to communicate between teams and base agencies, and 3 or 4 local area channels used for operations around the silos and Launch Control Centers. As a rule, these systems are located in the 148-150 Mhz range, FM-Narrow Mode, and are unscrambled. Here are some of the frequencies for the bases:
The UHF Aircraft Band in the 300-380Mhz band used to be busy with coded message traffic Between Command Posts, Looking Glass, and other nuclear triad agencies, especially on 311.0 and 321.0 Mhz. However, much of that traffic has now moved to secure satellite networks, although some channels still get occasional test messages sent.