The general rule is that you can’t study while on benefits except it is approved by Vinnumálastofnun.
If you want to take a course that will help you in your job-search you can apply for a study-grant. Vinnumálastofnun pays 50% of the course fee, but never more than 70.000 kr per year. If you are a member of a Labour union you can apply for a grant there also.
For Icelandic-classes Vinnumálastofnun pays 100%.
To apply for a study grant you come to a counsellor before the course starts.
If you want to study on a University level you have to sign a study contract with a counsellor and you can only take up to 10 ECTS per semester. On a high-school level you also have to sign a study contract and can only take 9 F- units at night school or distance learning.
Please talk to a counsellor if you are thinking about studying.
Sometimes Vinnumálastofnun offers courses but you can get more information about that from a counsellor.