If you've noticed a slight leak of oil from the rear axle of your Series Minx, you might be thinking of replacing the nearest oil seal.  The seal at the front of the differential is relatively simple to replace, but the hub seals can't be replaced without removing the hub – a very difficult operation.

Before resigning yourself to any work, there's one very simple thing you should do – check whether the breather hole is clear.  If it's clogged, pressure will build up as the oil and air in the axle casing become warm and expand during normal driving.  There comes a point at which one of the 3 oil seals can no longer contain the pressure, and a small amount of oil is expelled.

Section G of the Rootes workshop manual for Minx Series I to V has the following advice, with identical words for both the Spiral Bevel and the Hypoid types of differential :

The breather hole is 3/32" (2.38 mm.) in diameter and is drilled directly on top of the right-hand side of the sleeve of the axle casing 14.2" (360.1 mm.) from the centre of the banjo casing.  The breather hole should be kept clear at all times.

Use the standard chassis jack and remove the right side rear wheel.  Then you will easily be able to see the top of the axle casing.  This is better than using a pit or hoist to get under the car.  Scrape the mud off the axle casing to locate the hole.
Rather than using a piece of wire to push the dirt inside, turn a 3/32" drill bit in there to remove the dirt.  The pictures below show a drill bit in the hole.

Rootes produced a better-designed breather for the Minx & Gazelle Series VI.  Their workshop manual has the following advice at the end of section G :

The breather is housed under a domed cap situated on top of the axle casing to the right of the differential housing.

This cap seems to be made of thin pressed steel.  It's a loose fit to enable air to get past, but is crimped over to prevent it from falling off.  The likelihood of this version becoming clogged seems fairly slim, but give it a wiggle every time you do a service.