2015 Macbook Air Update

Your 2015 MacBook Air can upgrade to any available OS including the most recent one Big Sur.

I would recommend upgrading to macOS High Sierra first before upgrading to either

Mojave, Catalina or Big Sur.

The App Store Updates page only informs you of updates available for the OS you are running.

Installing a newer OS is called an upgrade.

For macOS High Sierra, macOS Mojave and macOS Catalina,

Open Safari and click on this link, 

How to get old versions of macOS – Apple Support

Go to the Download OS section and click on the OS you want to install.

Before upgrading make a full back up of your Mac.

Check that third party apps you use have updates available for the new OS.

https://roaringapps.com

Check that your printers/ scanners etc have updated drivers and software available for the new OS.

Connect your mac to your router via cable rather than rely on WiFi.

Do not use your mac while it is downloading

and do not use any other devices that share the same internet connection.

If the App Store says ‘Open’ then it is detecting a previous download of the install app

go to the Applications folder locate Install macOS ******.app.

Move it to the Trash, empty the Trash.

Restart, click on the link again, the App Store should say ‘Get' or 'Download'.

If you want to install Big Sur, open Safari and click on the link below.

How to upgrade to macOS Big Sur - Apple Support

If you are considering upgrading to Catalina or Big Sur be warned they do not support 32bit apps.

Your Mac is probably running 32bit apps and these will need to be updated to 64bit apps,

some updates may be free, some may be paid. Use this app to discover which apps you have

are 32bit, https://www.stclairsoft.com/Go64/ then contact the developers of the software to see

if they are offering 64bit updates.

If you are using Safari, then Safari is too outdated since Safari hasn't received any updates in years and the Internet has changed a lot since then. You will need to use a third party browser such as Google Chrome or Vivaldi (unfortunately Firefox is only supported on macOS 10.12+).

Since you have a 2015 Mac you can upgrade macOS to a later version where Safari is still being maintained. Your Mac can run the just released macOS 11.1 Big Sur, but you will need to make sure any third party apps are compatible with Big Sur and update those apps before upgrading to Big Sur. Both Catalina and Big Sur no longer support 32 bit apps so if you have any third party 32 bit apps installed that cannot be upgraded to 64 bit versions, then you will need to uninstall those apps before upgrading to Catalina or Big Sur.

32-bit app compatibility with macOS High Sierra 10.13.4 and later - Apple Support

Keep in mind if you have MS Office, that it will no longer be compatible with newer versions of macOS so you will need to purchase a newer version or use other third party alternatives. The same thing may apply for other third party apps.

You can find links to various macOS installers within this Apple article which also includes instructions for creating a bootable macOS USB installer (I highly recommend creating a bootable macOS USB installer so you have more options if something goes wrong):

https://support.apple.com/en-us/HT201372

Make sure to have a good backup before attempting to upgrade macOS. You may need it if something goes wrong during the upgrade or if you find out you don't like the new OS or you find an app no longer compatible.

You should always have frequent & regular backups since it is impossible to recover accidentally deleted data from an SSD plus an SSD can fail at any time without any warning signs.