On indented BoM's you would assign the Serial number in WO-I (Finish Production) and then issue those subs to the higher level assemblies.
 
SO Serial assignment is ONLY designed to deal with the finished item.

If you are talking about R type parts, you could use the lot number program to assign serial numbers .  Then when you issue them to a work order it would ask for that number and it would be recorded in DBA for that part.

Lorne,
Since we only use serial numbers for finished product I wasn't aware that DBA asks for serial number for R types when they are issued to a work order.  It sounded like, from your previous answer that DBA only assigns serial numbers to finished product.   And it also sounded like jrchilds was not seeing any control for items in lower levels of a BOM.  Wrong interpretation on my part.  I guess I need more coffee this morning;  we also use lot controlled products, and I realize lot control is completely different from serial control.  My suggestion was not a very good one!

Lorne,  
Well, now I remember where I was coming from-  for an R type part you could use a lot number ( after all they are both just numbers) and SO would prompt for the lot number (you could even rename it "serial" on the SO and Inv).  
But when receiving the part you would have to enter a unique number for each item, one by one, which would be a nightmare if you are receiving many of the same part. As  you said, a ton of work, and worth shuddering over.
However I got the impression that we are talking about  issuing a part  to a work order as part of the BOM and then trying to see the serial/lot number listed on the SO/Inv. I don't think DBA could do that with either the serial number for a finished part or lot number for an R type.
Cathy