When You Have Bad Links Impacting Your Google Rankings, Which Do You Remove?

I see this all the time, a forum thread, where a webmaster knows his rankings are suffering in Google because he was hit by Penguin because he has a lot of really bad, manipulative links. A WebmasterWorld thread sums up the issues a webmaster in this predicament is in.(1) They hired an SEO company (2) That SEO company ranked them well for years (3) Then Penguin smashed their links (4) They no longer rank as well (5) They are upset with the SEO company (6) They need to figure out how to rank again (7) Removing the links are the only option (8) But removing links that were the result of their initial good rankings won't help them rank immediately
In this thread, the site owner sums it up as:

1) What is the sure proof way to make sure a link is 100% bad? 2) I don't want to remove all links cause I am worried my site will drop even more. I'm sure there are some semi-good links that might be helping.
3) After submitting disavow file, typically how long does it take to recover? We have two sites, one seems to be under penguin and panda updates and the other received a manual penalty for certain bad links for certain terms.

It is sad, indeed. But you need to disavow the links, that is for sure. Those links are not helping you and they are now hurting you. Remove the hurt. Then get people to link to you because they want to link to you.
But which links should you remove? Which links are actually hurting you. That is the hard question. One SEO offered his advice:

the best advice I think I can give is to disavow the "obviously bad" links, but keep the ones you think are "grey" or "borderline" and see if you recover -- Basically, in your situation, meaning you don't "know" what's good and what's bad for sure, I'd "start with the obviously bad" and then "keep going" if necessary.

Of course, there are tools, like Link Detox, Majestic SEO, AHREFs, Moz and others. But we are assuming you have the tools already or you manually go through all your links within Google Webmaster Tools. And when you disavow, make sure to disavow on the domain level.
Forum discussion at WebmasterWorld.