I have macOS High Sierra 10.13 and the App Store says I have no software updates to install and I am having trouble with some recently installed apps that say my OS is out of date. Help

10.13.6 is as far as 10.13 can be updated, but there are upgrades if your Mac is new enough...

macOS Mojave 10.14.6 - Technical Specifications

General Requirements

OS X 10.8 or later

2GB of memory

12.5GB of available storage (OS X El Capitan 10.11.5 or later)*

Some features require an Apple ID; terms apply.

Some features require a compatible Internet service provider; fees may apply.

Mac Hardware Requirements

For details about your Mac model, click the Apple icon at the top left of your screen and choose About This Mac. These Mac models are compatible with macOS Mojave:

MacBook (Early 2015 or newer)

MacBook Air (Mid 2012 or newer)

MacBook Pro (Mid 2012 or newer)

Mac mini (Late 2012 or newer)

iMac (Late 2012 or newer)

iMac Pro (2017)

Mac Pro (Late 2013; Mid 2010 and Mid 2012 models with recommended Metal-capable graphics cards)

Don't use FireFox for these links…

https://itunes.apple.com/us/app/macos-mojave/id1398502828?mt=12

MacOS Catalina 10.15.7 - Technical Specifications

General Requirements

OS X 10.9 or later

4GB of memory

12.5GB of available storage (OS X El Capitan 10.11.5 or later)*

Some features require an Apple ID; terms apply.

Some features require a compatible internet service provider; fees may apply.

Mac Hardware Requirements

For details about your Mac model, click the Apple icon at the top left of your screen and choose About This Mac. These Mac models are compatible with macOS Catalina:

MacBook (Early 2015 or newer)

MacBook Air (Mid 2012 or newer)

MacBook Pro (Mid 2012 or newer)

Mac mini (Late 2012 or newer)

iMac (Late 2012 or newer)

iMac Pro (2017)

Mac Pro (Late 2013 or newer)

https://support.apple.com/en-us/HT210190

Don't use FireFox for these links…

https://apps.apple.com/gb/app/macos-catalina/id1466841314?ls=1&mt=12

How to upgrade to macOS Catalina – Apple Support

And caution on 10.15 or higher, to see which Apps will no longer work in 10.15 get Go64...

https://www.stclairsoft.com/Go64/

macOS Big Sur - Technical Specifications

General Requirements

OS X 10.9 or later

4GB of memory

35.5GB available storage on macOS Sierra or later*

Some features require an Apple ID; terms apply.

Some features require a compatible internet service provider; fees may apply.

Mac Hardware Requirements

For details about your Mac model, click the Apple icon at the top left of your screen and choose About This Mac. These Mac models are compatible with macOS Big Sur:

MacBook (2015 or later)

MacBook Air (2013 or later)

MacBook Pro (Late 2013 or later)

Mac mini (2014 or later)

iMac (2014 or later)

iMac Pro (2017 or later)

Mac Pro (2013 or later)

Learn about compatibility and how to identify your specific Mac model

Don't use FireFox or WaterFox for this link. it'll mess it up...

https://apps.apple.com/us/app/macos-big-sur/id1526878132?ls=1&mt=12

And caution on 10.15 or higher, to see which Apps will no longer work in 10.15 get Go64...

https://www.stclairsoft.com/Go64/

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.

Which OS you can upgrade to will be governed by which model and year your mac is.

Click on the  top left of your screen click on About This Mac to find out the year of your mac.

For Mojave, Catalina or Big Sur,

Open Safari and click on this link, 

How to get older versions of macOS

Go to Check Compatibility to see if your mac qualifies for the OS upgrade you want.

Then go to the Use Safari to download macOS 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 or are able to upgrade to Monterey open Safari and click on the link below,

How to upgrade to macOS Monterey – Apple Support (UK)

If you are considering upgrading to Catalina, Big Sur or Monterey 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 free 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.

Supernatural2 Said:

"I have macOS High Sierra 10.13 and the App Store says I have no software updates to install and I am having trouble with some recently installed apps that say my OS is out of date. Help Why won't my Mac download the latest OS?"

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Verify Compatibility:

What's the Model of your Mac?:

1. Go to: Apple Menu
2. Select: About This Mac
3. View: Second line-down

See if it is mentioned here: macOS Monterey is Compatible with these Computers - Apple Support

If it is not mentioned in that link, then installing it is just not an option. :(