Make it OPEN to ALL.
The Schengen system is for visa applications and associated intelligence. Surely we should provide read only access to everyone, then perhaps we might be able to spot when we are employing an illegal or when we come across someone "iffy" using an alias.
As far as I am concerned the more the merrier, can we link it to the NHS database so that we don't provide free healthcare to illegals too. Not just that, I have sponsored people to come to the UK and have been responsible for their healthcare costs, yet on three occasions they had to have medical treatment and I was not asked to pay a penny. So Schengen V3.1 can extend the database to sponsors and red flag those who have been involved in dodgy applications (I know they don't do this because I have made a few myself!!).
If we can add fingerprints to the database and then open it up to employers to verify their foreign workers against the database so we can avoid those nasty 10k per employee fines.
Oh and although it sounds obvious how about the border control take fingerprints and Retina scans on Non-EU nationals as they come into the country (making a digital copy of their passport at the same time) so that even if they bin them we can repatriate them even when the illegals bin them.
I had to provide fingerprints and retina scan on a recent visit to the US, I was actually quicker than passport control elsewhere in the EU going through the blue channel.
Hey remember that case last week of the "poorly qualified" foreign Doctor who killed a patient, can we use the Schengen system for that too, because we would never have allowed him to practice medicine here if he had come directly but another EU state was and once he got their OK we were forced to allow him to practice here.
Now you have to guess whether I am serious!!