Love it when I do a quick search for a QGIS plugin and find exactly what I want - today I wanted an easy way for a non-GIS specialist to find a UK postcode in QGIS. The ZoomToPostcode plugin (https://plugins.qgis.org/plugins/ZoomToPostcode/#plugin-details) does exactly what I want!
Thanks to @geospacedman for pointing out that the postcode data in this plugin is rather out of date.
I've just realised that you can use the built-in search bar in QGIS to search for UK postcodes using the Nominatim geocoder, which supports UK postcodes, by typing "> SO17 1BJ" or similar in the box. That's ">" and then a space, and then the postcode.