if you have a spare working 24/4 pin psu you can get a 24 to 6 pin adapter to test motherboard.  the green light on Dell proprietary psu is not guarantee that it is working.  unfortunately without a verified working psu it is hard to be sure whether it is psu or mobo that prevents POST.
I have a 3668 which had been running fine. I just added a new WIFI adapter in a PCI/E slot (the first time any new cards ever added to this system). It seated fine but when I reconnected power, I got a strong green solid LED on the power supply but the system wouldn't light the power button or power up no matter how long I held that button. Strangely enough however, all of the HD's started spinning up on their own and the CD-ROM drive would load/unload normally. All of the drives were running even though the power was off and neither fan was running. I removed the new WIFI card from the slot so that the system was as it had been previously but there was no change in the symptoms. I then tried replacing the CMOS battery and again, no change in symptoms. I then tried disconnecting all of the drives and again, no change in the symptoms. Finally, I tried removing each RAM DIMM and again, no  change in the symptoms. At no point during any of this testing or power attempts do I get anything but a bright, steady green light on the power supply, which to me indicates no problem with the power supply, I simply can't find any explanation for the symptoms or test results I am getting. The most confusing symptom is the fact that the HD's start spinning while power connected but off. I have tried all of the suggested Dell "no power" tests without success. Does anyone have any suggestions or ideas on what I can try next? Could I have a bad MB even though the LED stays green? Thanks in advance for any help.
Inspiron 3668 - Won't Power Up (Yet Drives Start Spinning When Power Connected) 
- Bad Motherboard?