Fragrance is consistently at the top of allergen lists, and usually in the top 5.
Even if you don't see "perfume" in an ingredient list, the product could still have fragrances or masking fragrances (scents that don't smell "perfume-y" but cover up the odors of other ingredients). They could be written in their chemical names like benzyl alcohol, balsam of peru, geraniol or cinnamic alcohol. In lieu of shopping with a chemist in tow, take a whiff. If the product smells nice it's probably got a perfume. If it smells bland it's probably got a masking fragrance. If it smells like a lab, like chemicals, it's probably fragrance-free.
Lots of preservatives are allergens. Quaternium-15, parabens (methylparaben, propylparaben, etc.) are top allergens. Try to steer clear of them altogether. CAUTION: many products achieve "preservative-free" status by piling on fragrances that have preservative properties.
Dyes are easy to spot: first, it's close to impossible to get a very bright color from mineral pigments, second, dyes are written as a color followed by a number and usually the word "lake", e.g. Yellow 6 Lake, Red 22 Lake or Blue 1 Lake. At VMV, some of our lip color, eyeshadows and blushes contain dyes - if you are allergic to dyes, choose from our wide selection of dye-free shades. All our foundations and powders are 100% dye-free.
Other well-known red flags are propylene glycol, rubber (common in cosmetic sponges), propolis (from bees wax) and tea tree oil