When should a receiver randomize across actions in a signaling game?
I think this cannot happen with risk averse senders, risk neutral receiver, and $A$ rich enough. 

For example, and to stick to the canonical signaling model, suppose that $A$ is the positive real line and senders' utility $u$ is increasing in $a$ while receiver's have linear utility decreasing in $a$. 

(Admittedly, this is only a partial answer as the framework is much less general that the one in your question, so it might not be satisfactory to you. I still provide an argument in case you were ok with these assumptions)

To derive a contradiction, suppose that at an equilibrium $\sigma^m_R(a') &gt; 0$ and $\sigma^m_R(a'') &gt; 0$ for some $a' \neq a'' \in A$. Let

$$a''' \equiv \frac{\sigma^m_R(a')}{\sigma^m_R(a') + \sigma^m_R(a'') }  a' + \frac{\sigma^m_R(a'')}{\sigma^m_R(a') + \sigma^m_R(a'') } a''.$$

By risk aversion

$$ u[ a''' ] &gt;  \frac{\sigma^m_R(a')}{\sigma^m_R(a') + \sigma^m_R(a'') } u(a') + \frac{\sigma^m_R(a'')}{\sigma^m_R(a') + \sigma^m_R(a'') } u(a'').$$
$$ [\sigma^m_R(a') + \sigma^m_R(a'')] u( a''' ) &gt;  \sigma^m_R(a') u(a') + \sigma^m_R(a'') u(a'').$$

Under some continuity assumption, there must also exist

$$ a '''' &lt; a''' $$ 

such that

$$ [\sigma^m_R(a') + \sigma^m_R(a'')] u( a'''' ) =  \sigma^m_R(a') u(a') + \sigma^m_R(a'') u(a'').$$

So consider $\sigma^m_R{'}$ constructed in the following way


$\sigma^m_R{'}(a') = \sigma^m_R{'}(a'') = 0$, 
$\sigma^m_R{'}(a'''') = \sigma^m_R(a'''') + [\sigma^m_R(a') + \sigma^m_R(a'')]$
For all other $\tilde{a}$, $\sigma^m_R{'}(\tilde{a}) = \sigma^m_R(\tilde{a})$


Receivers would prefer $\sigma^m_R{'}$ over $\sigma^m_R$ if it did not alter the signals sent by the senders, because it involves lower expected compensations. But by construction, senders are indifferent between $\sigma^m_R{'}$ and $\sigma^m_R$, so they should send the same signals as in $\sigma^m_R$.  Thus $\sigma^m_R$ cannot be an equilibrium which shows that we cannot have two different actions played with positive probability at an equilibrium.