Radar Warning Receiver systems (RWRs) used by military aircraft are omnidirectional. However, they do not warn of incoming missiles (unless that missile is at least semi-actively radar-guided, like an AMRAAM). What they warn of are various patterns of radar activity, including a radar lock (which is a constant, high-intensity scanning of the aircraft's position, more than just a "blip" as the antenna passes by on a wide-angle scan pattern).
If another aircraft has a radar lock on you, you should be expecting an incoming missile of some kind. However, IR missiles (AIM-9 Sidewinder, AA-11 Archer, MICA IR, ASRAAM) do not emit any EMR that indicates they're incoming; they use a passive FLIR sensor to identify and track the heat source they were told to kill (they don't even require a radar lock; the seeker can be cued to a pilot's helmet, or it can be "uncaged" and will lock on to the most significant heat source in front of it). Passive radar-guided missiles like the old AIM-7 Sparrow also do not emit their own radar; they track the radar returns from the firing aircraft. Only semi-active and active radar-homing missiles emit their own radar noise that can be used by the RWR to know one is on the way. A few older passive missile systems used radio control to guide the missile to the target, and RWRs can also detect these "sideband" radio signals as a clue a missile has been launched.
More modern Missile Approach Warning Systems (MAWS) have additional features that can help detect and track incoming missiles of multiple types. The biggest clue of a missile launch is a new heat signature, especially in the direction of an incoming radar lock; a suite of IR sensors can be used by MAWS to put 2 and 2 together and warn you of the missile launch.