My favorite lock format is the Axis lock. I understand all the possible concerns about it being complex, and therefore prone to failure ... but meanwhile, I haven't seen it. It is as reliable as any of the simpler lock formats, as strong if not stronger, does a better job of keeping the blade closed, and is ambidextrous. It is the one to beat, for me.
I love the integral/mono/whatever lock format as well. Being simple is a virtue -- although again, being more complex has not put the axis lock at any reliability advantage versus the integral, as far as I can tell. The integral lock is also very strong and smooth. Slight downmarks versus the axis due to the fact that it's not ambidextrous.
One other factor that hasn't been covered by anyone yet, is the restrictions the various lock formats make on the handle design. Integral locks are typically solid titanium handles, with some minor room for aesthetic flairs like inlays, grooves, and anodizing. But by and large, you're limited to a solid titanium handle, and the slightly heavier weight that that implies.
One exception is the Elishewitz integral lock, which in theory combines the best of what an integral has to offer for reliability, and the greater freedom of handle design that liner locks offer. I don't know how well this format works in practice, but it certainly seems promising.
The Axis lock does not limit the handle materials as much as the integral locks does, which means lighter weight materials and formats can be supplied. However, the Axis lock does seem to require a minimum thickness. The thickness isn't very noticeable on a bigger folder, where it's expected anyway, but on smaller folders, it can be noticeable.
Joe