Do all planets have a van Allen radiation belt? Earth has the van Allen radiation belts. Which other planets have one? Can it be determined if distant planets in other solar systems have one?
Mars does not have a magnetic field of any strength and so does not have a van Allen belt (and this is a serious problem for potential human exploration of the planet). Similarly, neither Venus nor Mercury (nor the Moon, which is arguably part of a two-planet system with the Earth) have a van Allen belt.

The gaseous giants do, however, have radiation belts which are analogous to Earth's van Allen belt.

Spotting aurorae on distant exoplanets would be an indication of a strong magnetic field and thus of the likelihood of a radiation belt. I am not aware that such aurorae have been detected but distant aurorae have been seen around a brown dwarf - a substellar object somewhat bigger than a planet (see http://www.theguardian.com/science/2015/jul/29/astronomers-find-aurora-a-million-times-brighter-than-the-northern-lights).

Alternatively, the radio emissions from high energy electrons trapped in planetary radiation belts might be detectable - this was how Jupiter's radiation belt was discovered around 60 years ago. Work seems to be underway to detect such radiation: https://skaoffice.atlassian.net/wiki/download/attachments/22183971/PoS-exopla-AASKA14.pdf?version=1&amp;modificationDate=1403003959989&amp;api=v2