The filters with anti-symmetrical impulse response all have a zero at $z=1$ (i.e. frequency 0). So if you need to implement a high-pass filter or derivative-like filter (or even band-pass), then you must go for types 3 and 4.
Similarly, if your filter is a low-pass type, then types 1 and 2 apply.
So, this depends on the type of filter you need to design, and not on which is more common.
Then, there is also a difference between types 1 and 3 vs. 2 and 4 in terms of phase response. There will be a extra $e^{j\theta/2}$ between the two types. Even if you don't care about the actual delay introduced, this half-sample difference can be important in terms of convergence in some cases of high-pass filters (the extra phase can make your frequency response continuous at $\theta = \pi$, therefore providing much faster convergence and a need for fewer coefficients).
In terms of implementation, all of the 4 types can be implemented efficiently without repeating the same coefficients twice.
You need, of course, the whole M-sized delay line. But instead of multiplying each of the tap outputs by its own coefficient, you first add (or subtract) the two corresponding outputs and then multiply only once by the coefficient.
For example, if impulse response is $h[n] = a \delta[n] + b \delta[n-1] + a \delta[n-2]$ (type 1 filter), instead of implementing $y[n] = a x[n] + b x[n-1] + a x[n-2]$, you make it $y[n] = a (x[n] + x[n-2]) + b x[n-1]$.