Can one take rebirth as an insect? I have always wondered if insects belongs to the animal realm in Buddhism, and I have yet to read stories about rebirth as an insect, such as a butterfly, a mosquito, or a praying mantis.
According to Buddhism, rebirth can take one of four types:


Opapatika - spontaneous rebirth
Samsedaja - born or arisen from moisture
Andhaja - born in an egg, oviparous
Jalabuja born from a womb, viviparous


Most insects belong to the andhaja group, I think. But yes, as others have mentioned, insects are considered a potential realm of rebirth.

Examples from the texts:

Queen Upari (Ubbari)


  Persons favored by fortune enjoying privileges of wealth, family, education, rank, physical beauty etc., are prone to exhibit haughtiness in their dealings with others. Shrouded in their own vanity and self-esteem, they become neglectful in their performance of meritorious deeds. Humility plays no part in their make-up. The Blessed One had taught in the Cūḷakammavibhaṅga sutta that such vain-glorious, haughty persons are liable to land up in lowly inferior re-births. On the other hand, unpretentious persons who show humility and pay reverential respects to those deserving of homage will be reborn in noble families.
  
  Queen Upari of our story was extremely beautiful and being the chief queen of the ruling monarch was of very high status in life. She had her head turned by these pre-eminent qualities and looked down with contempt on those she should have shown her respects. For such unwholesome attitudes and actions, it may be presumed she was reborn a lowly female cowdung-beetle. On hearing this account of rebirth of his beloved queen as a female beetle, King Assaka promptly rejected it, saying "I don't believe it."
  
  The hermit replied, "I can show you the female beetle and make her talk too." The king said, "All right. Please do and make her talk too." The hermit using his supernatural powers of abhiññā made a vow for both the male and female beetles to make their appearance before the king.
  
  When the male and female beetles emerged from the heap of cowdung into the presence of the king, the hermit said, "Oh King, the female beetle which is following from behind was your chief queen Upari devi. Having abandoned you, she is now trailing the male cowdung-beetle wherever it goes. Oh King, have a good look at the female beetle who was lately your chief queen Upari."
  
  The king refused to believe the hermit. "I can't believe that such an intelligent being as my queen Upari was reborn as this female beetle", said the king.
  
  True, for those who do not quite believe in the laws of kamma and its resultant effect, who do not understand the principles of conditionality or causal relationship, as explained in Paṭiccasa-muppāda, it would be difficult to accept that a being of the human world should have gone down so low as to become a mere beetle. Even in these days of sāsanā when Buddha's teachings are widely prevalent, there are some people holding the view that "when man dies, he cannot descend into an existence inferior to that of a human being". So it is not surprising that during the dark ages when Buddha's dispensations were yet unheard of, such stories of incarnation were received with scepticism.
  
  http://static.sirimangalo.org/mahasi/Dhammacakkappavattana%20Sutta.htm


Ven. Tissa


  Tissa.-An Elder of Sāvatthi. He once received a length of coarse cloth as a gift and handed it to his sister to be made into a robe. She had the cloth pounded and spun into fine yarn and made of it a soft robe-cloth. At first Tissa would not accept it but was prevailed upon to do so and had it made into a soft robe by skilled robe-makers. He died on the night it was finished and, as a result of his fancy for it, was reborn as a louse in the robe. After his death, the monks wished to divide the robe but the louse started shouting. The Buddha, hearing this by his power of divine audience, asked the monks to lay the robe aside for seven days. At the end of that period, the louse was reborn in the Tusita world. DhA.iii.341ff.
  
  http://www.palikanon.com/english/pali_names/t/tissa.htm