Is angle modulation spectrum always discrete or this property only holds for periodic modulating signal?
You will see lines in the spectrum of a phase-modulated signal if there is some periodicity in the modulating signal. If the modulating signal is not periodic, then the power spectrum of the modulated signal will be smooth.
As an example: consider a BPSK signal:
If the bits used to modulate the waveform are equally likely to be a one or a zero and are uncorrelated with one another, then there is no periodicity in the modulating signal. The power spectrum of a BPSK signal with independent equally-likely bits has a $sinc^2$ shape with no discrete lines.
If you then changed the data bits to add some repetitive framing data, for instance, causing a pattern of invariant bits in the modulating signal, you will begin to see lines appear due to the deterministic periodicity in the modulated signal.
In practice, one often makes an effort to ensure that bits are equally distributed to ones and zeros without repetitive patterns. Adding a source encoder (e.g. data compression) to your system can help to remove any repetitiveness in your data and whiten the transmitted bitstream.
One caveat to the above, courtesy of Dilip Sarwate's comment below, is duplicated here to make it more visible:
However, modulating signals such as $\cos(\omega_1 t)+\cos(\omega_2 t)$ where $\omega_1 / \omega_2$ is an irrational number (i.e. no harmonic relationship exists) are not periodic but have lines in the spectrum at $\omega_c \pm m \omega_1$, $\omega_c \pm m \omega_2$, and $\omega_c \pm m \omega_1 \pm n \omega_2$ plus a lot of clutter (smooth spectrum) in between the lines. Here $\omega_c$ denotes the carrier frequency. There are also similar lines (and clutter) near the harmonics of the carrier frequency.