signal analysis - Should I ever pick the continuous wavelet transform over the discrete one? DWT vs CWT vs STFT

Of course I mean in terms of numerical algorithms.

I'm reading various papers on the wavelet transform and why it's better than the short-time Fourier one. The reason that's cited more often is that the WT detects transients of known shape better than the STFT. In other sources I read that the DWT is essentially a faster version of the CWT. But so far all the MATLAB algorithms I've seen for scaleograms and transient detection have been using the CWT. Is there any specific reason for this?

In general, when should I code a CWT and when - a DWT?

Thanks in advance.