What are the different ways to find the envelope of a speech signal?
The notion of envelope is an ill-defined concept that describes the smooth upper and lower boundaries of more or less oscillating signals. 



The most classical technique rectifies the signal (take the absolute value) and low-pass filters the result. Of course, results are not unique. 

Other techniques identify local extrema and fit them with low-order functions like polynomials or splines. This is used for instance in some implementations of the Empirical Mode Decomposition.

The Hilbert transform sometimes offers a more solid alternative. Roughly speaking, suppose the signal can be written as the product of an envelope times an oscillation: $s(t) = e(t).o(t)$. Then, with other technical conditions, if the spectra of $e$ and $o$ are well separated, $H(s(t)) =e(t).H(o(t))$, where $H$ is the Hilbert transform. This is a loose transcription of the Bedrosian identity. 

If $o(t)$ is a fine pass-band oscillations (the purest being a sine or a cosine), and $e(t)$ is lower in frequency, the Hilbert transform can separate them. This is why for speech, the signal is often decomposed first through a multirate filter bank. 

All these techniques are often post-filtered in practice.

More recent approaches try to better define the regularity or smoothness of the envelope, using optimization techniques.