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Should I remove bad links e.g. 0 Page Rank, 0 Trust linking to old site that are being 301 redirected to new site?
Would I get rid of these links using Google's Disavow links?
We've only improved in rankings so far but I'm worried that the old links will only come back to haunt us.
Should I remove bad links e.g. 0 Page Rank, 0 Trust linking to old site that are being 301 redirected to new site?
Yes, you should. 301 redirect means Google will glue old domains with all it's parameters with the new domain. So, if your old domain is or will be affected by bad links in future, it will affect your new domain too.
Would I get rid of these links using Google's Disavow links?
Yes, you may. But, at first, I would suggest you to try to delete links physically by contacting webmasters and asking them to delete links. After you make everything you can and bad links will be present somewhere, you should export them to the tool "Google Disavow Links".
Good luck in your fighting with bad links.
Yes it will hurt your new domain because 301 will pass on the whole link juice either negative or positive to your new domain. You will need to submit disavow too for the old website not the new one.
