In addition to Google Webmaster Tools, Majestic Seo has a horde of bots that people volunteer to run. It crawls quite frequently so has a rather accurate view of your backlinks schema. There is also Open Site Explorer, but that seems to index less. If you want a neat way to check literal source for your links, NerdyData has a decent pool, although not as much as suggestions above.
Other than that, assuming you use Cpanel, an accurate way to find backlinks people are actually clicking through is using the Awstats module for the domain(s). Log into Cpanel then find the link for "Awstats". If its not there, ask your host to enable stats for your account, or if you have VPS, get into WHM and change the package/features for the account. When in Awstats, look on the sidebar for "Referring Sites".
Between the 5, this should give a decent scope of backlinks.
A reason why your backlinks may be ignored: If you find that Google no longer cares about your footer backlinks, well thats because you're more or less spamming your link. If your dev company has nothing to do "content-wise" with your clients (a la LSI), this would fall under the umbrella of "poor quality backlinks". Little by little Google may start to de-rank you because of this circa 2003 tactic. Also, no client wants your tags all over their site, especially if you don't pay them, so removing them is win win.
So what are proper backlinks in an LSI era? Well stick to sites within your keyword schema/scope. Forums, comments, tutorials, guest blogging, open source projects, etc are all great places to legitimately build a backlink pool that will actually work. Our favorite is the forum approach - simply answering other peoples questions, guessing what would be a hot guide, and including your backlink in signature. Google likes these more because its relevant content<->content.