There is a major split between English dialects where the pronunciation of the word bird contains an [ɹ] sound (e.g., western Canada, central and western U.S., northern England) and those where it does not (e.g., southern England, Australian, parts of the north-east and southern U.S.). In dialects which do not use an r sound in bird , the vowel between the [b] and the [d] is traditionally transcribed as [ə] or [ɜ].
Dialects which do use an [ɹ] sound in bird tend to use nothing but. In normal western Canadian or American speech, the period of time between the [b] and the [d] will be entirely occupied by an [ɹ] sound, and there will be no other vowel in the word. The [ɹ] is acting as the core of the syllable in [bɹd], a privilege which is usually reserved for vowels. A vertical line diacritic is used to mark those occasions where [ɹ] has a special vowel-like role in a syllable. The usual transcription bird is therefore [bɹ̩d]. (Sometimes you'll run across the symbols [ə˞] or [ɜ˞] or even the sequence [əɹ] used instead of [ɹ̩].)