If I need a surrogate, I would either use an IDENTITY column or a GUID column depending on the need for global uniqueness.
If there is a natural primary key, or the primary key is defined as a unique combination of other foreign keys, then I typically do not have an IDENTITY, nor do I use it as the primary key.
There is an exception, which is snapshot configuration tables which I am tracking with an audit trigger. In this case, there is usually a logical "primary key" (usually date of the snapshot and natural key of the row - like a cost center or gl account number for which the row is a configuration record), but instead of using the natural "primary key" as the primary key, I add an IDENTITY and make that the primary key and make a unique index or constraint on the date and natural key. Although theoretically the date and natural key shouldn't change, in these tables, if a user does that instead of adding a new row and deleting the old row, I want the audit (which reflects a change to a row identified by its primary key) to really reflect a change in the row - not the disappearance of a key and the appearance of a new one.