Dragon heartstring(s)
Dragon heartstring is a powerful wand with a lot of magical “heft”. It is not the core you want for subtlety, but for sheer power it is definitely the best. Although it is the most common core among Dark Wizards, Dark Wizards are most certainly not their most common users. Dragon heartstrings are by far the most common wand core amongst Slytherins, but their power often bonds to Gryffindors and Ravenclaws as well. However, they tend to overwhelm the archetypal Hufflepuff personality.

Phoenix tail feather(s)
Phoenix tail feather is a popular wand core due to its versatility and power. Its main strength lies in Defense Against the Dark Arts, although its adaptability can wrench it to hexes and jinxes if need be. As with the dragon heartstring core, the phoenix core is common amongst Light Wizards, but its users are not necessarily Light Wizards. This core may specifically impede Dark spells, so it is not common amongst Slytherins. However, it is by far the most common Gryffindor wand core, and is not unusual amongst Ravenclaws and Hufflepuffs.

Unicorn hair(s)
Unicorn hair is a more subtle wand, but it is quite compatible with Charms and Transfiguration. It is also hands-down the best core for healing, as it picks up some of the healing capabilities of unicorn blood. Unicorn hair has a reputation of picking gentler or more cerebral users, so it is common amongst Hufflepuffs and Ravenclaws. More laid-back Gryffindors and subtler Slytherins may find themselves with a Unicorn hair wand.

Exotics:
Acromantula web
If you're thinking about having a wand made with Acromantula web, you're probably a fan of having your wand confiscated by the authorities. Using a wand with this core has been illegal in Britain since 1782, after it was discovered that the wielder of an Acromantula web wand has particular ability with Dark magics, especially the Imperius curse. There are certain diplomatic exceptions, as it is a traditional core for Asian wands, but even those are temporary, and many wizard diplomats on long-term assignments find themselves compelled to procure replacement wands for their stay. 

Augurey tail feather
Augureys, or Irish phoenixes, were once associated with powerful Dark wands, as their cries were thought to signify an upcoming death. However, they were in reality never a strong Dark core, and were more accurately a powerful core for Divinations. Misunderstood students may find themselves bonded to an augurey wand, although these wands are altogether quite rare.

Basilisk skin
Basilisk wands are incredibly rare, as the beasts are rare to begin with and hard to kill. Due to the rarity, they often are passed down from generation to generation, so basilisk-core wands are either the heirloom of Slytherin-type Pureblood families or reforged wands from family cores. The occasional new basilisk wand will almost always bond to a Parselmouth or budding Dark Wizard. Very little good comes out of wielders of basilisk wands.

Billywig stinger(s)
Billywig stingers are not common in Britain, but are occasionally imported from Australia, the native habitat of the Billywig. Billywig wands bond almost exclusively to light-hearted pranksters, and are extremely capricious- at one moment it will produce the strongest Cheering Charm in the school, but another time it will object to being used as a potion stirrer and siphon up hours of work without so much as a by-your-leave. When they do bond to a witch or wizard, they tend to be of Hufflepuff or Gryffindor.

Boomslang venom
Boomslang venom, whether crystallized or in a rarer liquid core, provides a small boost to jinxes and hexes thanks to its venomous qualities. However, when a wandmaker undertakes the dangerous task of working with the raw venom, it is generally with the aim of creating a powerful Transfiguration wand. Whether or not the advantages outweigh the risks is not generally agreed upon in wandmaking circles. 

Chimera scale fragment
Although chimera scales are magically powerful, they are extremely rare in modern wandcraft. This is not out of any concern for safety, as they are generally considered no more stubborn than hippogriff feathers, and are more stable than Erumpent hide. The fact of the matter is that there are more recorded basilisk slayings in the past fifty years than there are chimera slayings in all of recorded history. This one slaying occurred in Greece over two millenia ago, so what scales were harvested at that time have been degraded, broken, and dispersed. 

Today, they are only found as parts of heirloom cores, and even then, all such cores are a more common core (often dragon heartstring) with a tiny fragment of scale embedded. Chimera wands are most common in Greece and the Balkans, although as they were circulated through the Mediterranean and former Roman Empire they are found throughout Europe. These wands are prized for their raw power, although they are difficult to control.

Demiguise hair(s)
Demiguise hairs were long considered to not have enough oomph to make a proper wand, but with the advent of multiple cores they have gained favor for their strength in Transfiguration and the subtle arts. When combined with a stronger wand core they make potent wands, however, on their own they can be rather one-dimensional and difficult to use for anything but Transfiguration. They have found favor in students of all Houses, although they may be slightly rarer among the open Hufflepuffs.

Doxy wing(s)
Doxy wings, like the creatures they come from, can be unmanageable and mean-spirited. They are second only to basilisk wands in their abilities with the Dark Arts, and as such these rare wands are most often found in the hands of stubborn Slytherins without the familial connection to obtain a basilisk core.

Erumpent hide
There is a very good reason this is an exotic- Erumpent hide wands are extremely dangerous, and don't take well to high levels of magic or sharp impacts. They may add a 'punch' to spells when combined with a gentler core, but most wandmakers refuse to work with it completely due to the danger it poses to maker and wielder.

Fairy wing(s)
This core makes for a light, airy wand, and is the absolute best for Charms. They also signify a connection to the mystic, so these wands, despite their relative rarity, are used by nearly half of known witches and wizards with the Sight. Despite their astounding strength in Charms, they are merely average in Defense Against the Dark Arts and Transfiguration, and will often fail at hexes altogether. As such, they’re not commonly found in Slytherin House, but are common in Hufflepuff. Ravenclaws and Gryffindors may be drawn to this core, although they tend to react better to blends of fairy wings and unicorn hair.

Fwooper feather(s)
Fwooper feather wands are said to be a mark of ill omen for the wizards they bond to, as, like the birds they come from, they are rumored to slowly drive their wielder mad. Despite their poor reputation, they do well with Charms and Care of Magical Creatures. However, they have a near-inability to castQuietus. They are commonly combined with another feather core, such as the phoenix for health or the hippogriff for stability. 

Hippogriff feather(s)
Hippogriffs are noble animals with a reputation for not taking a slight. These wands require constant respect, and if the wielder does not give it, they can watch its formerly stable and versatile magic backfire on it. It is not the strongest core, but it is one of the most adaptable. These wands are most common amongst Gryffindors, but they are rare overall.

Kelpie hair(s)
Kelpie hairs are incredibly tempermental cores, explaining their rarity. They were once common in Celtic wandmaking, however, the import of demiguise hairs has resulted in them falling out of favor. They have similar qualities to demiguise hair, and are powerful Transfiguration cores when they don't backfire spectacularly.

Veela hair(s)
Veela wands are temperamental like the creatures they come from, and are considered too volatile for a decent wand core in many circles. However, some wizards, particularly those with Veela blood, enjoy the boost it gives to outdoorsy magics, divinations, and Charms. The veela’s inherent intelligence makes finding these wands among the non-Veela blooded most common in Ravenclaw.