In allene (C$$_3$$H$$_4$$), the type(s) of hybridization of the carbon atoms is (are):

Allene has the molecular formula C₃H₄ and its structure is H₂C=C=CH₂. This means there are three carbon atoms: a central carbon atom connected by double bonds to two terminal carbon atoms, and each terminal carbon is bonded to two hydrogen atoms.

To determine the hybridization of each carbon atom, we count the number of electron domains around the atom. An electron domain can be a bond (single, double, or triple counts as one domain) or a lone pair. The hybridization depends on the total electron domains: 2 domains correspond to sp hybridization, 3 domains to sp² hybridization, and 4 domains to sp³ hybridization.

First, consider the central carbon atom. It is involved in two double bonds (one with each terminal carbon). Each double bond is counted as one electron domain. Since there are no lone pairs on the central carbon, the total electron domains are 2. Therefore, the central carbon atom is sp hybridized.

Next, consider a terminal carbon atom (for example, the left one). It has one double bond with the central carbon (which counts as one domain) and two single bonds with two hydrogen atoms (each single bond counts as one domain). Thus, the total electron domains are 3. There are no lone pairs, so the terminal carbon atom is sp² hybridized. The same applies to the other terminal carbon atom.

So, in allene, the central carbon atom is sp hybridized, and both terminal carbon atoms are sp² hybridized. Therefore, the types of hybridization present are sp and sp².

Now, examining the options:

A. sp and sp³ — Incorrect, as there are no sp³ hybridized carbon atoms.

B. sp² and sp — Correct, as it matches the hybridization found.

C. only sp² — Incorrect, because the central carbon is sp hybridized, not sp².

D. sp² and sp³ — Incorrect, as there are no sp³ hybridized carbon atoms.

Hence, the correct answer is Option B.