Compassionately, many Christians tell their little children that Fluffy will be in heaven, with every good kitten toy, waiting for them to join them in heaven in the distant future (old age) when they ascend to heaven. This is a little white lie to a kid so the kid won't cry.
The Christian bible has many verses about dogs, which describe them as filthy animals that will not ascend to heaven.
In the old days, animals were working animals (bark to alarm the village, herd sheep, etc). Now, pets are beloved members of our household. Some would rather spend time with Fluffy than dreaded Aunt Mildred. However, this doesn't affect pet acceptance to heaven.
Trailblazer mentioned that souls are energy and energy is neither created nor destroyed, but continues to exist forever. I believe that souls are made of psychic energy, and have the ability to transcend time and penetrate any material. In fact, souls are able to penetrate any material except the vortexes of psychic energy that surround heaven (keeps psychic energy out) and hell (painfully keeps psychic energy in). Gravity doesn't seem to have an effect on psychic energy, but time dilates at relativistic speeds and in strong gravitational fields, according to the laws of relativity; hence time slows for God which makes the universe 6,000 years old in God's years and 13.8 billion years old in human years. However, there is nothing to suggest that animals have souls (so they might not possess souls). One might ponder why pets sometimes sense quakes in advance (perhaps they hear micro-fractures before a quake?).
I think that souls are only put into humans because only they have minds capable of using them to fullest capacity.
However, there are many religions that believe in reincarnation, and they often speak of animal spirit guides (some are delicious and nutritious....so not a good idea for everyone to seek out theirs).
Dog passages of the bible: Proverbs 26.11, Exodus 11:7, Exodus 22.31, Judges 7.6, 1 Samuel 17:43, 1Samuel 24:14, 2 Samuel 3.8, 2Samuel 9.8, 2Samuel 16:9, 1Kings 14:11, 1Kings 16:4, 1Kings 21:19, 1Kings 21:23, 1Kings 21:24, 1Kings 22:38 (They washed the chariot at a pool in Samaria (where the prostitutes bathed), and the dogs licked up his blood, as the word of the Lord had declared)., 2Kings 8:13 2Kings 9:10, 2Kings 9:36, Job 18:11, Job 30:1, Psalm 22:16, Psalm 22:20, Psalm 59:6, Psalm 59:14, Psalm 68:23 (that your feet may wade in the blood of your foes, while the tongues of your dogs have their share)."
Do Dogs Go to Heaven?
Link above: "Heaven is for those who are saved."..."God created animals to be ruled over by humans. Animals are a gift and responsibility for men to steward, they aren’t equals to men in the eyes of God."
Of course, this opinion doesn't seem to be consistent with the idea of evolution (animals are not created for mankind, but, rather, mankind is descended from apes).
Link above shows that Isaiah 11:11 "The wolf shall dwell with the lamb, and the leopard shall lie down with the young goat and the calf and the lion and the fattened calf together; and a little child shall lead them." I don't know if this is a description of heaven. If so, it is possible that animals can get into heaven.
From the opera Pocahontas: "have you ever heard the wolf cry to the blue corn moon, or asked the grinning bobcat why he grins" (my theory is that the bobcat ate the wolf...grin...burp).