One that has been in the news again recently is the authenticity of the Turin Shroud, which is currently on display and is expected to attract millions of visitors once more. This linen shroud is believed by many Christians to be the cloth that covered Jesus Christ after his crucifixion.
In 1988, it seemed that this was disproved, as carbon dating on a small strip of the shroud dated it from 1260-1390, which suggested that it was no more than a medieval fake. However, the controversy surrounding the authenticity of this artefact has remained, especially since several scientists have cast doubts over the reliability of this evidence.
This leads me nicely onto where the shroud can be used in science lessons. The original tests in 1988 provide a nice context for the uses of carbon-dating. (This animation is a good student-friendly explanation of how carbon dating works). You could also use this website to calculate the age of the shroud. If you tell students that the sample still contained 92% of its original C-14, the calculator on the site will place the age of the cloth (or more precisely, the age of the linen plants that were used to make the cloth) to be 600 years old. The site also contains a half-life graph which shows where these numbers came from.
You could also look at the reliability of these results. This PowerPoint presentation can be used as a starter for GCSE students, and asks them how the reliability of the results from the 1988 carbon-dating results could be improved. The issue lies with two sources of contamination: Prior to the 19th century, the shroud used to be held up by its top-corners by bishops when being displayed, and from smoke during a fire in 1532.
Many scientists would love to get their hands on the shroud and carbon-date pieces from many different areas of the cloth, but the Church do not want the relic damaged. So, it looks like, for now at least, that this is one mystery that will be left unsolved.