Can angels speak to us? This is perhaps slightly related to this question, and specifically this answer which points out that "the worship of angels [is] false doctrine ... praying to angels arguably may not be worship, [but] in my mind the line is far too fine."

So we have, perhaps, a caution to pray/speak to angels.  But can angels, under normal circumstances, speak to us?  I am excluding, for the sake of this question, situations in which angels are physically manifest, and speak to us verbally.

What evidence exists in the Bible that angels either can or cannot speak to us this way?

And if the abilities/rules governing God-fearing angels is distinct from fallen angels/demons, what are the differences?
Technically, "angel" (from the Greek Angelos) is a job function, not genus and species.  Angelos means messenger - they deliver messages.  If the messenger is unable to communicate, it doesn't really seem up to the job of delivering a message.

The angels recorded in Scripture all clearly had the ability to converse with the people to whom they were sent - whether to Abraham, Manoah (Samson's father), Mary, Joseph, Zechariah, etc...  In the cases where the angel's identity was hidden, there was nothing to suggest they appeared anything other than human, along with the concomitant powers of humans.

As to whether or not Cherubim and Seraphim (the heavenly creatures typically given these jobs) have the ability to communicate when not in the presence of humans, most Protestants would say, there would be only speculation. Historically, the doctrine of God's omnipresence and omniscience is what gives us the ability to communicate with God in any place - there is nothing to suggest that heavenly creatures - angelic or demonic - possess this attribute.

As svidgen points out, however, there are interpretations of Revelation 8 that Catholics would interpret as giving Angels the ability to hear prayers.