ספירת העומר - S&P - London [EA]
ספירת העומר - ברכת המצוה - S&P - London [JC]
In the London S&P tradition the Hazan sings תהלים קכא [EA] and/or תהלים סז [EA] which, on special occasions, may be sung the congregation and/or choir, as here and here.
During the ספירת העומר period, the following melody is used for לכה דודי [MF], as well as for שיר המעלות [MF, music SF] in the Western אשכנזי tradition.
A version of this לכה דודי embellished by Lewandowski and others is sung in many communities in place of the original:
לכה דודי - ספירה [KAJC] {a little slower than usual}
לכה דודי - ספירה [HB] {Dutch variant}
לכה דודי - ספירה [CG]
יגדל - ספירת העומר - Netherlands
Additionally, there is a specific chant for a segment of the ברכה following the morning שמע of שבת:
על הראשונים [MF]
There is a custom to refrain from listening to instrumental music during the עומר period. For an article I have written on the subject, see here. A shiur from R' Eli Pick on the same subject: Music during ספירה
There is a widespread custom to declare one's intent before doing a מצוה, which is particularly popular with regards to ספירת העומר specifically. A shiur from R' Dovid Sulzbacher on the subject: הנני מוכן ומזומן