This Modified is using a K&N...
This Modified is using a K&N cotton gauze filter. Notice the amount of dirt and mud that has impacted the filter. This is a bad thing, as there is an opportunity for the mud and dirt clods to damage the filter. Once that happens you have to replace the filter. If it happens at the start of a race, you have the potential of running a race with effectively no filter.
These are free-flowing, low-restriction air filters. They do an admirable job of balancing airflow and have the ability to clean the air before it enters the engine. They are not as good at taking out the fine dust and grit in the air, but the oil added to the cotton fiber filter helps trap some of it. These are very robust and work well in applications where the filter may come in direct contact with debris. Cotton gauze filters are designed to be washed and reused. While they are very robust, they're not indestructible.
Your third choice is a foam filter. These were first used in the motorcycle industry, specifically the dirt bike market. Foam filters are manufactured using an open-cell foam construction, and they are oiled with special oil that is very sticky. They look very much like a sponge but are not as dense. These filters offer very good filtration and reasonable airflow. They are washable and are quite durable if reasonable care is used. The downside is that they can be damaged very easily and are prone to tearing if not handled with care. Foam filters need to be in an airbox for protection. These filters have never really gained any traction in the automotive field, but not because they do not work well. There has simply never been a great deal of product made for the auto industry.
Even after you have made the decision about the type of filter to run, you can make one more choice and install a shield over your filter. Many racers started using women's nylons to add more protection to the element, but this has evolved into a very specific segment of the filter business. Outerwears(tm) offers a product that fits on the outside of the filter to give it more protection. As an added bonus, the Outerwears(tm) products are treated with water repellant, which helps prevent water from entering your engine.
So what is the correct selection? As usual, it is important to use data to answer the questions that we need to ask as racers. If you race on a dirt track and the filter is covered with mud and/or other debris after the race, you may opt for a more resilient design, one that is not as affected by water and is more structurally sound-the cotton gauze filter. If you still want to run a paper filter, it may be in your best interest to build an airbox or a filter box to protect the filter from the elements and the track. Often, you do not have the extra room to build an elaborate filter box.
Not running a filter is not an option at all. The filter should be regarded as a tool that prevents damage to your engine, and in today's racing world engines are not considered low-cost consumables. They represent a very significant segment of your racing budget, and as such, should be given the same treatment and care as the rest of the car. The engine is going to wear out, and it is your job to make sure that it is fed a clean, filtered quantity of air so that the process is not accelerated. Selecting a quality filter for your specific application is much easier and less expensive than rebuilding your engine.