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In Edit Mode I click on Randomize under Mesh Tools to randomise the position of vertices on a mesh:

enter image description here

What I've noticed is that this randomises the position on all axes. Is there any way to do the same randomisation of vertices but constraining it to one or more axes (locally or globally)? A mockup of the desired output is shown below in which the vertices are randomised along the global x axis.

enter image description here

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You can use Proportional Editing set to a random falloff.

Basically, select the vertices you want affected, press Ctrl + I, to invert the selection, and then H to hide those vertices. Now, select a vertex of the ones left, press O, to turn Proportional Editing on, and, in the 3D view header, set the falloff to random:

Random falloff

Then, press G, to move, followed by one of X, Y or Z, either once or twice, to limit to the axis you want. Now, use the scroll wheel to enlarge the affected area so that all the vertices are affected and move the mouse until you get the results you want. Then, press Alt + H to show all hidden vertices again.

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I don't think there's any tool to achieve that directly under edit mode. You could however achieve similar results with a displace modifier. If I am not mistaken, displace can move vertex along the local surface normal, then use some texture to simulate randomization.

Edit: Alternatively if you want to move all verts in a single direction you could use the Proportional Edit with Random editing fallout

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