Ultimately, it doesn’t really matter. Both round and square bench dogs will hold effectively.
After a short time of working with both shapes though I must say that I soon came to favour the round ones. Many, many bench builds later and that opinion still stands.
As someone who’s pleasure comes from working wood by hand I have to see the workbench as the absolute epicentre of my workshop so my case for round dogs is built from a purely functional point of view.
All dogs have a flat face for clamping against and whether they’re round or square you can add a slight angle to this to aid clamping. The way I see it is there’s nothing a square dog can do that a round dog can’t, but there’s plenty a round dog can do that a square dog can’t.
Ability to rotate.
This is really the key point for me as I’ve found a big difference in the ability to clamp out of square boards with round and square bench dogs.
A square dog’s face is set square to the bench top which will suit most situations when making furniture but in practice I often rely on my dogs rotating to be able to hold work securely. If I’m preparing rough timber by hand then the ends are usually somewhat out of square from where I’ve quickly cut the boards by hand without marking. When clamping boards like this to flatten their faces my round dogs will adjust themselves to hold secure as I tighten the vice, with square dogs and heavy planing the work would slip out.
There are many more times where I’ve had to clamp slightly angled or even curved work as part of a build and square dogs are much less practical for this.
Multiple hole Use.
I find this point important because I use holdfasts a lot when I work. By using round bench dogs I can make sure that every hole is also suitable for a holdfast making the bench top versatile without littering it with all manner of holes.
I also like to use dogs and holdfasts along the front face of my bench whether its in a deep apron, board jack or the leg and I feel these areas lend themselves to round holes rather than square.
Ease of creating the holes.
In my experience round dog holes are easier to create than square ones. These can be done on a pillar drill or hand held drill using a guide – see this post. Square holes take a lot more effort either through brute force or in setting up a means to rout them. The routing depends on having a laminated top and making the cuts before gluing up, which can then be prone to a weaker glue line and fussy glue up.
If retrofitting a vice in to an existing bench then round holes are much easier to add in my opinion.
After saying all that I’m keen to add that these benefits are based on my personal way of working and may be less relevant for yourself. I can see that for someone using their tail vice mainly towards the later end of a project for smoothing perfectly square parts then square dogs may be preferable. I know a lot of people also prefer the look of a square dog so there’s certainly no clear cut answer only personal preference – it’s a bit like saying that tea is better than coffee or bitter better than lager!
And lets not forget that for many woodworkers a tail vice isn’t even considered necessary, so it’s best not to lose too much sleep over this one…