Question

What does the path of an object look like when it is in uniform motion?

Answer 1

Uniform motion is a motion in which an object covers equal distance in equal intervals of time and moves along a straight line.

Thus the path is a straight line in uniform motion
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Answer 2

Uniform motion:

Uniform motion is a form of motion when there is uniformity seen in the distance travelled with the relation to time. This means that distance travelled will be equal in time periodically.

So, the path followed does not play any role in determining a motion to be whether uniform or non-uniform, the factor of determination is just the equal distance travelling equal time periodically.  

So, the path can either be a straight line or it can be curved path travel or can also be a zigzag path. The direction does not play any role only the magnitude does.