I'm trying to find a report that can be created or customized to yield some basic information.

In psuedocode, what I'm trying to accomplish is this:

For Each Computer In Collection
For Each Required Software Patch
If Patch is Not Installed
Print Computer Name and Software Patch
End If
End For
End For

Are there any reports that exist to yield this information?

Give this a try. It is not exactly what you are looking for but it is a good start.

http://smsug.ca/blogs/garth_jones/archive/2009/02/25/patch-compliance-progression-report.aspx

Another :

http://www.systemcenterdudes.com/sccm-2012-software-update-reports/

try this...may be it will help you

https://ashusccm.wordpress.com/2015/07/17/patch-information-in-sccm-2012-tablesviews 

• Edited by Ashribad Rout 13 hours 27 minutes ago

try this...may be it will help you

https://ashusccm.wordpress.com/2015/07/17/patch-information-in-sccm-2012-tablesviews 

• Edited by Ashribad Rout Wednesday, August 19, 2015 6:23 PM

None of these really show required updates. They show software patches.

None of these show the big list I'm looking for...the reason for this is to identify trends within the collection.

The report at "\Monitoring\Overview\Reporting\Reports\Software Updates - B Deployment Management\Management 2 - Updates required but not deployed" gives a better indication of what I need, but it provides no visiblity to the computer names of the machines reported for each patch. If this report could be modified to show the list of computer names for every patch, that would be acceptable as well.

I've reached out to these people, I'm hopeful that they will have what I'm looking for.