To kick off with, how about a group of organisms capable of interbreeding and producing fertile offspring. Of course reproductive isolation may mean that separated components of a lineage may interbreed if they come into contact again after a period of isolation, but duration of separation will decide if these are phenotypic morphs or distinct species in their own right. Some orders of insects are more prone to subspecies recognition ... these could be speciation events that have not completed. There are some asexual species and a few parthenogentic ones, but on the whole, the reproductive (or biological species sensu Mayr) definition seems to work.
That is true. But what I am concerned about is the still followed morphological species. I have seen that many insect taxonomists erect new species based on a few structural characters. For example -presence of a spine. In what way such a structure leads to the evolution of a new species? Does it impart mating incompatibility?
Sometimes few characters are all there are to go on morphologically speaking, but if those few characters repeatedly and consistently appear in one population and not in another (perhaps geographically isolated), then there may be grounds for describing a new species. In the flies that I work on, spines on the ventral surface of the fore femur of males are certainly part of a mating display and possibly agonistic behaviour. This feature and others in the head (e.g. lateral expansion of the genae (cheeks), or of the head itself into "eye-stalks") are used in male-male stand-offs, the winner of which wins/keeps the harem. Development of bigger spines and wider head modifications therefore become pathways to genetic success.
The concept of specie or delimitation of species is not clear... There are several unifying concepts of what a specie is or how to delimitate them. Genetic concept, morphological concept, reproductive concept and a few more. This paper can help you to understand and study one unifying concept: De Queiroz, K. (2007) Species concepts and species delimitation. Systematic Biology, 56, 879–886.
I completely agree with Andrew, that reprodactive definition is best way for species delimitation, however this is difficult to use in every case, but still I think that this is the only way to get exact answer on the question.