1998 toyota corolla all three wires to the headlight are hot. I checked all the relays and the switch. The car has daytime running lights, but no headlights.
The wires may be shorted together, maybe in the connector to the headlamp. Have an electrician check the wires for shorts to other wires and to grounds. This is English to an electrician.
Bad ground?
The headlight circuitry is not as simple as, only, relays and fuses. There is a control module. Here is the wiring diagram for your electrician: http://www.autozone.com/shopping/repairGuide.htm?pageId=0900c152801a88d8 Click on the second Power Source (1998) to reveal the headlight wiring diagram. Click the maximize square, download and print.
The reason you see 12 volts on all the wires is because the system uses a switched ground lead to control the light circuit rather than switched power. The problem is most likely with the switch or wiring to it. The link to the drawing that Hellokit so nicely provided helps show what is going on with the circuit.