It depends on what failed. Most of the time, sudden "spectacular" failures (loud pops or sparks) are on the primary side and that type of failure rarely damages anything beyond the PSU since a blown FET or other power device on the primary side cannot send any more power to the secondary side after it has blown up.
What tends to fry components is progressive failure on the secondary side where dying filter caps let an increasingly large amount of ripple through. If nothing fails or shuts down the computer first, those ripples will eventually exceed the absolute maximum of some components and damage them. Excessive ripple will also slowly chip away at all other caps in the system and may eventually cause local regulators to malfunction...