The conducted emissions testing for MIL-STD-461 is similar to that of the DO-160 Conducted Emissions testing. For MIL-STD-461C you can use an EMI Analyzer along with a Current Monitoring Probe to verify the quality of your power lines. For pre-compliance testing, most current spectrum analyzers will work as well, as they provide the capability to program in your own limit lines. They will not work for compliance testing unless they have the required EMI filters built in.
Additionally, it is highly advised to use a transient limiter such as the Agilent 11947A in line with the input to your EMI/Spectrum Analyzer to prevent any damage to the high value instrument in the event of an unwanted power spike.
CE01 of MIL-STD-461C will require a seep between 30 Hz and 20 kHz and CE02 will be done between 20 kHz and 50 MHz.The Current Versions of MIL-STD-461E/F/G are done using LISNs instead of Current Probes: CE101 has narrowed the frequency requirements, covering 30 Hz to 10 KHz and CE102 10 kHz and 10 MHz.
Video Demo of MIL-STD-461