Which Kanban software (preferably Saas) would you recommend that has integration with GitHub or BitBucket?
I tried many and I truly recommend the best kanban tool I've ever tried so far: Kanban Tool.
We
use zapier.com to integrate github with Kanban Tool and it works pretty
well in our company. We added an "external link" to our card template.
As soon as someone commits any changes, he adds a link to the commit. To see the commit just click on the link icon on the card. And it works awesome!
I would also say that if your team is very small Trello might work for you.
Blossom is a Kanban tool that integrates with GitHub quite well.
Some other alternatives:
You can check out SwiftKanban. It is available on both SaaS as well as on-premise - and is integrated to over 30-35 tools, including Git - see the full list here.
Full Disclaimer: I am a product manager at SwiftKanban.