The physical angular velocity is angular displacement divided by time interval. The dimension of angular displacement is angle (A); the dimension of time interval is time (T). So the dimension of angular velocity is A/T.
Confusingly, what is commonly known as "angle" (e.g. in the SI) is the radian measure of the angle: if theta is an angle (dimension A), the corresponding SI "angle" is its radian measure, "theta" = theta/rad (dimension A/A = 1). The SI "radian" is the radian measure of 1 radian: "rad" = (1 rad)/rad = 1.
The radian measure of theta, theta/rad, is the number of radians in theta--i.e. a number (dimension 1). So what is called "angular velocity" (in the SI and almost everywhere else) is the number of radians swept out (dimension 1) divided by the time interval (dimension T): its dimension is 1/T. In other words, (SI) "angular velocity" is the actual physical angular velocity divided by one radian.
As an example, if theta = 20 pi rad (dimension A) and t = 10 s, the (actual physical) angular velocity is omega = (20 pi rad)/(10 s) = 2 pi rad/s. The SI "angular velocity is: "omega" = omega/rad = 2 pi (rad/rad)/s = 2 pi 1/s = 2 pi "rad"/s. [The SI helpfully explains that 1 can be replaced by (the SI) "rad" where appropriate (the quotation marks are not used).]
The well-known widespread confusion permeating this subject stems from the ubiquitous formula:
"theta" = (s/r)
where "theta" is said to be "measured in radians." What this actually means is that "theta" is the numerical value of theta when theta is expressed in radians: theta/rad--i.e. the radian measure of theta. So the formula is actually:
theta/rad = (s/r)
This is dimensionally consistent: dim(LHS) = A/A = 1; dim(RHS) = L/L = 1.
Rearranging for theta:
theta = (s/r) rad
Keeping rad EXPLICITLY in the formulas for rotational kinematics and dynamics dispels the confusion.