Sorry for the delay...
1. Trello
Trello is based on freemium business model. It is not open source.
Through many kanban based tools were already available, Trello simplified it for larger audience. It is worth to mention that we used Trello for years and here we have a good fan base for Joel Spolsky, et al.
When we used Trello, we came across some issues like frequent disconnects, can't use it in offline, can't access more than one card at a time, can't quickly add a card to different board, etc. We tried to bridge those gaps in Restyaboad. Our priority is/was speed and performance.
2. Taskboard
Taskboard is a nice simple open source software. According to its intro "It's not trying to be the next Trello". Anyhow, you may check their demo and see if it suits your requirement.
3. taiga
taiga is python based open source software. It is a PMS with kanban like UI. It is not directly like Trello.
In our market research, many people want flexible kanban board. That's why we have also used Trello for longtime.
taiga also offer Trello like free hosting. You can still try them and see if it fits your requirement.
4. Kanboard
Kanboard is PHP based open source software. It doesn't have much Ajax; mostly rendered from server. They have many additional features. In our market research, those features may not be needed for the mass. That's why in Restyaboard, we're keeping it thin and working on App platform for easy extensibility. You can still check their demo and decide.
5. Wekan (libreboard)
Both are same. libreboard changed their name to Wekan. It is Meteor based open source software. They started it as end-to-end Trello clone and so received DMCA notice from Trello. They too have features that are not in Trello, like multiple select, etc. In our market research, those features are not that most wanted. Even as per our own experience with Trello, speed and offline support are most needed. If you want UI close to Trello, this might be the ideal choice. Here in Restyaboard, we're trying to position ours as better than Trello. Wekan also offers some hosting option. So, you can still try that and decide.
Having said, you may not get unbiased opinions in a product forum. We strongly believe that with our upcoming release, we'll have almost everything and you may also check our comparison page with trello
http://restya.com/board/comparison.html Some hosting companies are also contacting us to offer instant Restyaboard setup in their platform. So, we may sometime offer hosting options like taiga and Wekan.
You can still try demos of all these products and decide.