Memorial Prayers 1 - אב הרחמים

Following the Prayer for the Congregation and/or the Prayer for the Government, there is a מנהג in Eastern אשכנזי congregations to say a prayer in memory of martyrs. (Western אשכנזי congregations do so after אשרי, and that only twice a year: שבת חזון and the שבת before שבועות, where public memorial prayers are considered seasonally appropriate.) This prayer was probably written in response to the Crusades [1] (which would easily explain why ספרדי congregations don't say it at all, as they, by contrast, were experiencing the Golden Age around that juncture in history).

There are also specific joyous שבתות of the year where אב הרחמים is not said. While strictly speaking every congregation is entitled to establish their own מנהג, as there are not really any חיובים or איסורים one way or the other, nonetheless there are some basic guidelines given by the פוסקים, which we will present here:

1) We don't say אב הרחמים:

⦁ Wherever we don't say צדקתך צדק (itself a subject for its own article) by מנחה on שבת 

    ⦁ except תשעה באב

⦁ where there is a חתן present

⦁ where there is a ברית in shul 

    ⦁ except during ספירה, because the Crusades took place in that part of the year

        ⦁ unless ר"ח אייר is on שבת

        ⦁ Where ר"ח אב is on שבת is a מחלוקת מנהגים

    ⦁ on שבת מברכים (except during ספירה, as above)

2) We do say אב הרחמים all other weeks of the year (i.e. by default), including:

⦁ שבת מברכים חודש אב

⦁ The part of חודש ניסן that overlaps with ספירה

References:

רמא או"ח רפד, מ"א שם ס"ק ח, לבושי שרד שם ס"ק ט

[1] Hertz, Rabbi Herman J., The Authorized Daily Prayer Book (Bloch, 1951), pg. 511