If Software Update says that your Mac is up to date, then macOS and all of the apps it installs are up to date, including Safari, Messages, Mail, Music, Photos, FaceTime, Calendar, and Books. If you want to update apps downloaded from the App Store, use the App Store to get updates. Mac App Store is the simplest way to find and download apps for your Mac. To download apps from the Mac App Store, you need a Mac with OS X 10.6.6 or later. The App Store gives people around the world a safe and trusted place to discover apps that meet our high standards for privacy, security, and content.
If a blue account button appears at the top of the App Store, you might not be signed in. Tap it and sign in with your Apple ID. Learn what to do if you forgot your password.
After performing the updates the App Store no longer displays a 'checking for updates' spinner, and I can successfully install updates to third party applications, quit and relaunch the app store, etc. I will hazard a guess that the OS X 10.10.3 update, which I installed via the command line, fixed the issue with app store. For some history, the Mac App Store started handling system software updates starting from Mac OS X 10.8 through Mac OS X 10.13, and then the process was moved back to the System Preference Software Update control panel in MacOS Mojave 10.14 and Catalina 10.15 onward. If you’re running any release from macOS 10.13 to 10.9, you can upgrade to macOS Big Sur from the App Store. If you’re running Mountain Lion 10.8, you will need to upgrade to El Capitan 10.11 first. If you don’t have broadband access, you can upgrade your Mac at any Apple Store. Learn more about how to upgrade.
Learn what to do if you can't open the App Store.
You might need to have a payment method on file, even if the app that you want to download is free. See what payment method you have on file and make sure that it isn't expired, declined, or failed.
If you can't update apps and you see a message that says “your account is disabled in the App Store,” there might be a problem with your payment method. Contact Apple Support for assistance.
From the Home Screen, touch and hold the app. From the menu that appears, choose Prioritize Download.
Restart your iPhone or restart your iPad.
Then check to see if you can download or update apps.
If you still can't download or update apps on your iPhone or iPad, contact Apple Support.