It is not normally necessary for a client application to access this property, as the list is automatically created by the DicomConnection object when failures occur, and it is automatically passed on to the requesting application in the final status message.  It would, however, be necessary for an application to modify the contents if it wished to accomplish C-MOVE operations using the Send method instead of using SendImages.
The list of failed SOP instances is present as

DicomConnection.Failures.Attributes(0008,0058)

which will return a string array.

DicomConnection Object  | DicomConnection Members