On Holiday
Eikhah is chanted on this day of mourning.
Tisha B'Av
Eikhah is traditionally chanted at this time (or service) during Tisha b'Av.
Evening (ma'ariv)
Ecclesiastes (Kohelet) and Song of Songs (Shir HaShirim) are both chanted on this day of the holiday.
Shabbat of Hol HaMoed
M'gillat Esther may be chanted at these two times during the holiday of Purim.
Ma'ariv and Shacharit
In Eikhah, this entire chapter is one big detour.
Chapter 3
This Megillah is named for Purim's Jewish queen.
Esther
In Eikhah cantillation, these two disjunctive phrases sound almost exactly the same.
M'gillat Ruth is read on this day of the Hag.
Second
Many of the ta'amim phrases in Esther cantillation can also be found in this m'gillah.
Eikhah
M'gillat Esther allows for creative detouring in many places. The most detours are chanted to the ta'amim of this other m'gillah.
Eikhah
This is the m'gillah we chant on Sukkot.
Ecclesiastes (Kohelet)
Eikhah was reportedly written by this prophet.
Jeremiah
The Shalosh R'galim m'gillot are chanted at this service (time of day).
Shacharit
While there are many proscribed detours in M'gillat Esther, one is also encouraged to detour from the detours and do this with the cantillation.
Be creative or improvise
In M'gillat Esther, the names of Haman's sons are traditionally chanted in this manner.
Quickly (and in a single breath)
This is the m'gillah we chant on Pesach
Song of Songs (Shir HaShirim)
In each chapter in Eikhah the verses are all arranged alphabetically, except for this one.
Chapter 5.
Just like Eikhah, where the last verse is superseded by an earlier verse, this m'gillah also has a "false" ending.
Ecclesiastes (Kohelet)
The verse לַיְּהוּדִים, הָיְתָה אוֹרָה וְשִׂמְחָה, וְשָׂשֹׂן, וִיקָר, (For the Jews there was light and happiness, joy and glory) also appears in the liturgy for this weekly occasion.
Havdallah
Verses from Shir HaShirim (Song of Songs) are frequently used in this lifecycle ritual.
Wedding
These three m'gillot are all "Festival" m'gillot and have the same trope.
Song of Songs (Shir HaShirim), Ruth, & Ecclesiastes (Kohelet)
Because in Eikhah, like in the books of the Prophets, we try to end on an "up note", so we do this after we've chanted the last verse (verse #22) in Chapter 5.
We go back and repeat verse #21--
השיבנו יהוה אליך ונשובה חדש ימינו כקדם
The authorship of Shir HaShirim and Kohelet are attributed to this king.
Solomon
M'gillat Esther contains these two ta'amim which are not found in any of the other m'gillot.
Yerach ben Yomo and Karnei Fara
Eikhah cantillation can also be found in these other texts which are read/chanted around Tisha B'Av.
Haftarah, Shabbat before Tisha B'Av--Isaiah
Torah, Shabbat before Tisha B'Av--D'varim
Haftarah for Tisha B'Av--Jeremiah