How common are Levirate marriages today?

Question

Under Jewish law, if a woman's husband dies without giving her a child, the brother of the deceased husband has an obligation to give his wife a child. In the event that they don't wish to go through it, they can cancel out the process through chalitzah.

The question I have is how common of an issue is this today and does it still come up?

• Is the brother in law more more likely to push for a Levirate or the widow?
• In the event neither push for it, can a Rabbi push the issue for them?
• Is it one of those things where it always gets canceled through chalitzah and the process is simply a formality (no one really expects the two to have a child)

Answer 1

Levirate marriage today would seem to be very rare today. Reason being it's virtually never done in the Ashkenazi world and while more acceptable in the Sefardi, it seems to rarely ever takes place.

From the "History" section in the Wikipedia entry on Yibbum:

By Talmudic times the practice of levirate marriage was deemed secondary in preference to halizah by some of the rabbis, because of the brother's questionable intentions (see Bechoros 13a) indeed, to marry a brother's widow for her beauty was regarded by Abba Shaul as equivalent to incest (Yevamos 39b). Bar Kappara also recommends halizah (Yevamos 109a). A difference of opinion appears among the later authorities, with R' Yitzchak Alfasi, Maimonides, and the Spanish school generally upholding the custom, while Rabbeinu Tam and the Northern school prefer halizah (see Shulchan Aruch, Even Haezer 165:1-5). [...] Additionally, if the surviving brother is married, Ashkenazim, who follow the takkanah of Rabbeinu Gershom which abolishing polygamy, would be compelled to perform halizah.

Orthodox Jews in modern times have generally upheld the position of Rabbeinu Tam and perform halizah rather than yibbum. Yemenite Jews, though Orthodox, practiced yibbum until the en masse Aliyah of Jews to Eretz Israel in the last century ¹

However, even though Sefardim had more of a tradition to practice yibum, nowadays in Israel even Sefardim don't do yibum. A Torah Musings article titled "Levirate Marriage Today" gives the reason for this:

There was a ruling by both [Ashkenaz & Sefard] Chief Rabbis in 5710 (1950), that all residents of Israel, including immigrants from edot mizrach (eastern countries such as Yemen, Iran, Ira, and Syria), should no longer perform yibum [as this yeshiva.co article states "so it should not look like there are two Torahs"] (see R' Herzog's responsum on this, Heichal Yitzchak 1; Even HaEzer 5)

This isn't universally agreed upon however, as said Torah Musings article also quotes R' Ovadia Yosef who argues on this ruling and states that Sefardim are allowed to perform yibum in Israel.

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^{1 Goitein, S.D. (1983). Menahem Ben-Sasson (ed.). The Yemenites – History, Communal Organization, Spiritual Life (Selected Studies) (in Hebrew). Jerusalem: Ben-Zvi Institute. p. 306. OCLC 41272020}

Answer 2

On wikipedia, it says,

Orthodox Jews in modern times have generally upheld the position of Rabbeinu Tam and perform halizah rather than yibbum. Yemenite Jews, though orthodox, practised yibbum until the en masse Aliyah of Jews to Eretz Israel in the last century.

The relevant siman in the Tur is Even Ho'ezer 165, in which it shows that R' Yizchok ben Ya'akov al-Fasi and R' Moshe ben Maimon holds that yibbum takes priority over chalitzo, whilst R' Shlomo ben Yitzchok holds that chalitzo takes priority of yibbum but if the man and woman want to do yibbum we let them and are not concerned about them doing it not for the sake of the mitzvah, but R' Ya'akov ben Me'ir holds that chalitzo takes priority of yibbum in all cases.

As you can see here (pages 56-57), the Ashkenazi psak of compelling the man and woman to do chalitzo caused a special prenuptial chalitzo document to be made at the time of marriage, but as you can see there, it has pretty much fallen into disuse.