You must notify The Directorate of Labour immediately about all changes that occur to your personal circumstances which may impact your rights while you are receiving unemployment benefits. The same applies during the time you are not receiving payment due to penalties. If you do not notify of such changes, it may lead to a temporary or permanent suspension of payments. 

All changes must be reported via My Pages.  

You have an obligation to report the following:

• Studies a school certificate must be submitted.
• Incidental work must be reported at least the day before.
• Part-time jobs must be reported at least the day before. 
• Full-time employment You have to de-register from the unemployment registry. 
• Other income you receive alongside unemployment benefits such as rental income and other capital income, payments from pension funds, unions, health insurance and The Social Security Administration. 
• Establishment of a company (ehf., slf., sf.) / shares in a company. You must notify us when business operations start again after having been ceased temporarily, or if a third party is recruited at the same time as an owner receives unemployment benefits.
• Illness The Directorate of Labour can request a medical certificate
• Impaired work ability/incapacity for work A medical certificate must be submitted which confirms your capacity to work to day, as well and from which date and work capacity ratio.  
• Residing abroad without a U2 certificate must be reported at least the day before. 
• Changes to your telephone number and/or e-mail address.

In the video you can see how to notify of incidental work or income via My Pages: