From what I can tell this is an outdated practice in the web design industry.
"made by awesomedesign" links was pretty popular back in the day I guess.
For SEO they dont really help much, unless the site you made hits it big and starts bringing alot of link juice. And even then it might be seen as spammy by google or unrelated since the sites arent in the same niche.

Footer link is good for rankings but use it carefully anchor text must be generic.

It works perfectly but you must use your brand name (assuming that you have your main keyword in brand name) to rank safely. A single footer link is counted multiple times (with each page on the target site) and if you go out to target a specific keyword, it may bring penalty. (In case you get too many links from that particular website)

Hello I heard about footer before but don't know what it is. Can someone please explain to me what a footer is? Thanks

Build your site for humans and not search engine.

Having link to your site in the footer is just fine.

I think the best way to improve your SEO ranking is to make sure the links to your site are useful and helpful.
If you just put a link to your site without any context then I don't think you'll find it overly beneficial however if you add some context to it then it would hold more weight.

It would be even more beneficial to you if you had your link on a page that explained a little about your service but that depends on if that page would be of any benefit to the readers of the website they are on.

You wouldn't be sending people from dog breeding to a web development site as it wouldn't be of use to them.

Their's no problem to have your link on client site, but it should have good backlinks and the niche should be as yours. If your client site is new, then don't link with them until it has ranking for some other keywords.

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I also think that links in the footer still caries some value. I do not think that you will be penalized for leaving a link in the footer.

If the website you're creating, with your link in the footer, will go viral, then it will probably help your ranking s and your business as well. But if this isn't the case, I'd still say you can leave your link in the footer. It's a possibility to gain another client or two, you never know.