To do quick spot-checks of different domains, the way to go would be the on-demand backlink checker. This tool lets you quickly check the backlinks of any domain. 

Each backlink check uses one credit. Premium plans come with a monthly credit allowance, and if you run out, you can purchase more.

You can access the backlink checker by going to the Link Checker section:

If you're on a paid plan, make sure you click on "On Demand" so that you're using the Backlink Checker without the restrictions of the free version.

You can read more on how to quickly check the backlinks of a random domain here.

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