As well as following the advice of the event organisers to ensure you've got all the compulsory kit, you'll need to do your own research into likely race conditions to ensure you've got the right kit.
Shoes and kit for multi-day running
Equipment
Depending on the event, you may be sent a comprehensive list of compulsory equipment to wear or carry. This can vary from an anti-venom pump to a specific number of safety pins. Whether or not the event requires compulsory kit, do your own research into likely race conditions to ensure you've got the right kit.
When choosing kit for any type of multi-day event buy the smallest, lightest equipment you can afford
Most importantly you must know how to use your kit, having practiced with it time and time again in training. This will ensure you are fully aware of its capabilities, limitations, comfort factor and the effect this has on you.
One good tip when choosing kit for any type of multi-day event is try to buy the smallest, lightest equipment you can afford. However there are limits - you don't want to buy the world's smallest compass and subsequently find you can't follow that all important bearing!
Shoes
Your feet may well swell if you run a long way!
When you start upping your mileage, it is important to realise that you may require different size shoes. This not a science and everyone is different. However, in this country on a 40 miler, your feet may swell a little.
For races in this country such as the Compton 40 and the Thames meander 50, I found a shoe 1/2 a size bigger than my normal shoe size a must. In the Sahara desert, your feet will swell! During the MdS, I found that a shoe a full 2 sizes bigger was necessary.
Rucksacks
For races that require a rucksack and numerous pieces of equipment, find out what the majority use. There are a wide variety of specific running rucksacks and bumbags on the market. Fit is really key - try some on in the shop and then make sure you test them well, running with all the kit you'll need for the race.
Sleeping Bags
The race may invite unusual equipment. For sandy desert events, waterproof socks and home made sand gaiters are common
In events that are predominantly dry, down sleeping bags offer a greater warmth for their weight. If competing in the Highlands of the Scotland you may want something more suited to wet conditions.
As with all kit considerations, carry out detailed research into the specific needs of the environment you will be in - for example a sleeping bag suitable for the Moroccan desert will not be suitable for the colder Jordan desert.
Other
The race or the terrain may invite unusual and eccentric variations on equipment - the race organiser and previous competitors will be able to help here. For example for sandy desert events, a combination of waterproof socks and home made sand gaiters are common. In colder wetter climates, comparatively heavy, goretex lined multi-terrain trainers might be appropriate.