That total is nearly double the other prolific fraudsters and thieves in the top 10.
Second on the list was 681 convictions, followed by 614, then 613.
The top 10 is rounded out by someone with 492 convictions, including 191 public order offences and 105 against "justice procedures, government security and government operations".
The only other member of the top 10 whose criminal record is not dominated by fraud charges is "Offender 6". Of the 585 convictions, 353 are for theft and 167 for public disorder.
Sensible Sentencing Trust spokesman Garth McVicar said the names of the offenders should be made public in the interest of an open justice system.
"We would even go one step further - we believe all that information should be available on a public website.
"Our justice system is funded by the public and they say we have an open and honest justice system, but actually we don't - it's secret-squirrel stuff."