I've created a site that shows a few of the many amateur radio contacts that are currently occurring across the world.

Many many contacts take place every minute across the world. Some of these are logged on a network called the DX Cluster.

The site is ham.calum.org.

(A QSO means a contact in Q-code terminology)
If the QSO is on the HF bands, you can listen to the frequency on http://websdr.ewi.utwente.nl:8901/, an online SDR system based in the Netherlands. (This doesn't necessarily mean that you'll hear the contact. Perhaps it was logged after the contact finished, or perhaps the propagation from the stations to the Netherlands is poor/non-existent.

The mode of the contact is guessed from the frequency used. SSB = Voice, CW = Morse Code, narrowband = Digimodes (e.g. PSK31 or RTTY)

There is also a map, which shows any QSOs that include two Maidenhead locators. Most of the contacts with locator codes in them though are European/US amateurs, and mainly on the higher frequency bands.

The symbols inside square brackets in the notes say what type of propagation has been used.
AS = Aircraft scatter
AU/AUR = Aurora
AUE = Auroral Es
BS = Backscatter
EME = Earth-Moon-Earth/Moon bounce
ES = Es
F2 = F2 layer
FAI = Field Aligned Irregularities
ION = Iono-scatter
MS = Meteor scatter
RS = Rain scatter
SAT = Satellite
TEP = Trans equatorial propagation
TR = Tropo