Above: CSI's colonial-era Spanish-style מנורה (or חנוכיה, as they insist on calling it)
The first ברכה upon lighting the מנורה, made on the performance of the actual מצוה itself:
להדליק נר של חנוכה [MF]
The second ברכה upon lighting the מנורה thanks 'ה for making miracles on behalf of our ancestors:
שעשה נסים [MF]
The third ברכה is made on מצוות that occur infrequently where they are done for the first time that period of התחייבות:
שהחיינו [MF]
Here are some recordings of all three ברכות together:
ברכות הדלקה 1 - Netherlands [unidentified]
ברכות הדלקה 2 - Satmar [unidentified]
ברכות הדלקה 3 - Satmar [unidentified]
ברכות הדלקה 4 - Western אשכנזי (Germany?) [unidentified]
ברכות הדלקה 5 - Bukhoro [unidentified]
ברכות הדלקה 6 - Vizhnitz [unidentified]
ברכות הדלקה 7 - S&P? [unidentified, with choir]
ברכות הדלקה 8 - Morocco [unidentified]
As the candles are being lit, one recites a paragraph stating what the נרות represent, why they are being lit, and what they may otherwise be used for (hint: nothing):
הנרות הללו 1 [DIR]
הנרות הללו 2 [MF, composed LL]
הנרות הללו 3 [Lubavitch choir]
The above song was composed in a cantorial style largely foreign to Lubavitch. This was done to purposely lengthen its recitation so as to cause the Rebbe at the time to stay with his followers for a longer period after he finished lighting the מנורה.
הנרות הללו 4 [AAC]
At the completion of the lighting of the מנורה, the following Psalm is chanted antiphonously in a psalmody by the חזן and congregation (or, in this case, choir):
מזמור שיר חנוכת הבית לדוד [LSC, JC]
Here is all of the above in one recording:
complete מנורה service 2 [EAb]
Here is a selection of other parts of the liturgy that have special melodies for חנוכה:
ונשמח - מעריב [ADC]
ראו בנים [FS]
נשמת כל חי [FS]
אל ההודאות [FS]
שמחים בצאתם [FS]
ממצרים גאלתנו [FS]
נקדישך [FS]
הלל [FS]
הלל - אודך כי עניתני [FS]
כתר [FS]
על הנסים [FS]
Shiurim from R' Eli Pick on עניני חנוכה:
Though there is a near-universal custom to sing מעוז צור at this point, that song is themed on גלות וגאולה, having nothing more to do with חנוכה than with פסח or פורים.