Cats, cockroaches. While any allergies can be associated with and trigger asthma symptoms, there are 2 in particular which have been shown in the medical literature to be highly linked to asthma flares. Cats allergies are by far the most associated with asthma, followed by cockroach allergies (most common in inner-cities). Chlorine has also been associated with asthma, but it is not considered an allergy.
Inhaled allergens. Inhaled allergens are more likely to trigger asthma than food allergens. Animal danders, especially cat dander, are significant triggers as are house dust mites & cockroach. Mite particles enzymes may break down the epithelial barrier in the breathing tubes leading to additional allergic sensitization. The mold spore alternaria is thought to be the cause of thunderstorm-triggered asthma.