Singlet and triplet carbenes exhibit divergent reactivity.  
Triplet carbenes are diradicals, and participate in stepwise radical additions. Triplet carbene addition necessarily involves (at least one) intermediate with two unpaired electrons.  
Singlet carbenes can (and do) react as electrophiles, nucleophiles, or ambiphiles.  Their reactions are typically concerted and often cheletropic.  Singlet carbenes are typically electrophilic, unless they have a filled p orbital, in which case they can react as Lewis bases.  The Bamford-Stevens reaction gives carbenes in aprotic solvents and carbenium ions in protic ones.
The different mechanisms imply that singlet carbene additions are stereospecific but triplet carbene additions stereoselective. Methylene from diazomethane photolysis reacts with either cis- or trans-2-butene to give a single diastereomer of 1,2-dimethylcyclopropane: cis from cis and trans from trans.  Thus methylene is a singlet carbene; if it were triplet, the product would not depend on the starting alkene geometry.