From my observation, last minute inventory appears to be going to SQ's e-upgrade but not necessarily to awards and upgrades.

Initially, SQ introduced the 14 days cut-off time to decide on how many seats they wanna keep open for last minute commercial passengers and how many they can approve for awards/upgrades using miles. It appears that when someone cancels and award/upgrade on points, this sometimes can go into inventory.

Your last option is the Advantage Upgrade at check-in which should be possible for any seat unsold shortly before departure.