Radiated EMI is tricky if you want to roll your own. Conducted is easier.
But, there are probes for E and H fields that work quite well. I have a set of probes from Credence Technologies, Santa Cruz, CA that I have used for, I think, around 20 years.
There are probably others, this set is named ScanEM-C Model CTK015. Not very expensive. Can also be used without scope - LEDs show relative field strength.
A small open coil can be used to detect and probe EM leakage from gaps in shielding, points of high current, etc. Keep in mind that it will be uncalibrated. Start with ten turns and about 1cm diameter. You can make much smaller versions if required. With some experience, you'll quickly see how to use it.
Keep the wavelength in mind, but even HF can have small hotspots.
Around 100 MHz I'll take a length of BNC coax and chop one end off. Strip back the shield and then wire a small loop from the center conductor to the shield; I've used loops from 0.5" to 3" depending upon my frequency range.
Note that this really works better with a spectrum analyzer or a signal strength meter. The noise floor on an oscilloscope is much to high to see emissions that will cause you to fail FCC/CISPR emissions tests.
With all respect, gentlemen. Those techniques work for magnetic fields, but not necessarily for the E-field. And that may be all that the OP needs. But he also qualifies his question with "Broad range of frequencies" and I do not expect small passive loops to work well with standard oscilloscopes when frequencies go down to a MHz or less.
Even if he decides to order a set of commercial probes, he can still try winding a few coils while waiting for the package to arrive. The investment in winding and playing with some homebrew coils is about as close to zero as can be imagined.