Can you disconnect the power lines? What sort of current are you measuring- is it continuous or a burst? What accuracy? Since you want to measure them simultaneously you are going to want some cheap method
If you can disconnect the lines you can insert some device like a DVM (which will only measure and average or RMS current). If that is insufficient then you will need some kind of probe to hook up to an oscilloscope. And then there are resistors or hall effect devices and then measure the voltage. Do you want a complete result or can you make up/design the hardware yourself.
Oscilloscope manufacturers produce current probes that clamp around the wire so there is no need for disconnection. These probes can be expensive. There are other devices like those made by Hioki (
https://www.hioki.com/en/products/list/?category=17 ). Just google "Current Probes".
But let's get back to your "simultaneous" need. Shunt resistors would be the simplest, wired to an input multiplexer hooked up to the computer, BUT it depends on the voltage that you are measuring. For instance a possible approach might be the 570M-OEM from Sealevel Systems (http://www.sealevel.com/store/i-o/analog-i-o/rs-485-modbus-rtu/570m-oem-rs-485-modbus-rtu-interface-to-8-16-bit-a-d-8-isolated-inputs-8-form-c-relay-outputs-multifunction-module.html ) but it only allows for +/-15V analog inputs. For 20 inputs you would two of these.
There are other manufacturers of devices like this so we may be able to find something if you can give us a few more specifications.
Strictly speaking though, the measurements are not simultaneous, but although they are very close together they are sequential.
Another solution if you can make the hardware and depending on the currents could be to use the Allegro current sensor ICs (http://www.allegromicro.com/en/Products/Current-Sensor-ICs.aspx )
For what it's worth I did a series of blosg on different current measuring techniques
http://www.eetimes.com/author.asp?section_id=36&doc_id=1319944and
http://www.eetimes.com/author.asp?section_id=36&doc_id=1320111&page_number=1