The (inter)dental sounds dh and th in English are distinguished by all
speakers, but simple rules with few exceptions can accurately predict
which sound is used in any given word. Because grammatical category and
some etymology are needed for the rules, this may not be enough reason
to conclude that there is only one combined th-dh phoneme in English.
Yet there are very few minimally differing pairs.
The rules for prediction of th/dh, with examples, are as follows:
Adverbs derived from adjectives by suffixing -ly: thinly, thoroughly
Numerals
Cardinals: three, thirteen, thirty, thousand
Ordinals: third, thirteenth, thirtieth
Prepositions: through
Voiced [D]:
Adverbs (not made by suffixing -ly): thus, then, there
(EXCEPTION: voiceless when r follows: thrice)
Pronouns
Relative: that
Demonstrative: this, that, these, those
Personal: thee, they
Possessive: thy, thine, their, theirs
Conjunctions: than, though, thence
Definite article: the
In medial position:
Voiceless [T]:
Words of Greek origin: sympathy, mathematics, author, epithet.
But: rhythm (pronounced by some with [D]), vs. rhythmic!
Biblical names, mostly of Hebrew or Greek origin:
Methusalah, Golgotha, Bartholomew
Proper names (some also because they're from Greek):
Cynthia, Bertha, Dorothy, Timothy
Compounds and suffixed words: something, nothing, strengthen,
lengthen
(EXCEPTIONS: worthy, swarthy, Smithy (also with [T]), but earthy,
stealthy again with [T]).
Voiced [D]:
elsewhere: father, farther, gather, brother, brethren,
whether, rather.
(EXCEPTION: brothel)
(Note: brethren is one of the very rare cases where a voiced [D] and
an [r] touch in the same word)
In final position:
Voiceless [T]:
Nouns: moth, myth, length, growth, wreath, sheath
(Note: many of these words' plurals have [Dz], at least with some speakers)
(EXCEPTIONS: scythe, Hythe, booth, tithe)
Adjectives: loath, uncouth (BUT: see under final [D] too).