Several days ago my Dell OptiPlex 960 MT SATA RAID HDD PC would not boot into Windows 10. After a long unsuccessful odyssey of trying to repair whatever caused this boot-up issue I had a new SSD installed AND much of the data that was on the HDDs was "deleted" inadvertently and I have no backup either.

I connected one of the RAID drives to my fixed PC with a USB to SATA cable hoping to see what is on the HDD. Although I got the usual sound when one connects a new device, it doesn't show up.

I know I can send the HDDs away and pay several hundred dollars to possibly get my data back, I am hoping there is a less expensive solution.

I will greatly appreciate advice.

What RAID level was it?

Assuming (and hoping) it is RAID1, try scanning the drive with DMDE.
Possibly the "deletion" also deleted the MFT, and that's why connecting the drive directly doesn't show up.

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before You scan or do anythig else please make a 1:1 copy of the HDD.
DMDE is good to make a copy (image) to another HDD or to a file.
Than scan the copy

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I downloaded DMDE and selected Disk Images, clicked OK and Open Drive Image popup presented but the HDD connected to my PC with a USB / SATA cable is not seen. When I plug the USB in, the sound that signals a device was attached occurred but the HDD did not appear.

????

If you can connect all required members of the array to an internet enabled PC, there are many competent members here who can help, but you have to take into consideration the possibility that a failed boot could relate to some surface errors or could be early warning signs for a more serious failure for which remote work cannot handle potentially risking the data itself.

In an ideal situation, you would get this looked at by someone who can properly deal with it safely and professionally who can also deal with any more serious faults that may be present.