I believe the Halacha is that you're supposed to tear your clothes when seeing the Wailing Wall, just as one does when 'sitting Shiva'. Why don't most people?
There are quite a few ways in which people avoid tearing, though I guess most don't do it simply because they are unaware of the Halacha.
The Kotel is not the best place to tear - one should try and see the makom hamikdash (Temple Mount).
The Shulchan Aruch (OC 561:2) writes that when seeing ‘Yerushalayim in its destructive state’, one should say ‘Zion has become a desolate desert’ and tear keriah. Upon seeing the ruins of the Beis Hamikdash one should tear keriah (again) and say ‘Our house of holiness and glory in which our ancestors sang praise to You, and all that we hold precious has been destroyed.’
The Mishna Berura (and others) writes that Yerushalayim is only considered to be ‘in a destructive state’ when it’s under foreign rule. Thus, R’ Moshe Feinstein writes (Igros Moshe OC 4:70) that as Yerushalayim is under Jewish rule today, one need not tear keriah when seeing Yerushalayim. One does, however, upon seeing the Temple ruins.
While R’ Moshe Sternbuch writes (Teshuvos Vehanhagos 1:331) that one tears upon seeing the Dome of the Rock, others (Baer Heitev 561:5) write that it is ideal to view the Har Habayis itself. Ideally one should try to find a higher vantage point to accommodate all views.
As one doesn’t need to tear keriah if he has been within 30 days, some sell their shirt to a friend while others go to the Kotel on Shabbos or Friday afternoon (the ‘first time’) to avoid doing so. R’ Moshe Feinstein writes (Igros Moshe YD 3:52) that one still tears on Friday afternoon, however.
There are opinions that one does not need to tear kria now that Jerusalem is in our hands. For instance, R Shlomo Aviner (in his book Rabbenu Ha-Rav Tzvi Yehudah Ha-Cohain Kook, p. 158) writes
The Halachah rules that one must tear his garment when seeing the spot of the Temple in ruins (Moed Katan 26a and Shulchan Aruch Orach Chaim #561). Rav Yosef Karo, in the Beit Yosef, in discussing the obligation to rip one’s garment upon seeing the cities of Yehudah and Jerusalem in ruins, explained that we hold that "in ruins" means "under non-Jewish control." The Magen Avraham (#1) and Mishnah Berurah (#2) accepted this view. This means that even if there is a Jewish settlement in the Land of Israel but it is under non-Jewish control, it is still considered in ruins, and one must tear his garment upon seeing it.
Our Rabbi, Rav Tzvi Yehudah Ha-Cohain Kook, explained that the same applies for seeing the spot of the Temple in ruins, and just as "in ruins" means "under non-Jewish control" for the cities of Yehudah and Jerusalem, so too does "in ruins" mean "under non-Jewish control" for the Temple Mount.
Therefore after the famous call of "Har Ha-Bayit Be-Yadenu - the Temple Mount is in our hands" during the Six-Day War, he ruled that there is no longer an obligation to tear one’s garment when seeing the Temple Mount, even though the Temple is still destroyed.
Our Rabbi explained that it is possible to claim that since there is no Temple, one must tear their garment. One must understand, however, why we are prevented from fulfilling the Divine Commandment of "Make for Me a Temple" (Shemot 25:8). The Temple Mount is in our hands and our inability to build the Temple is not due to "exile." We are in control, but are prevented from building the Temple due to halachic and political reasons. These are our reasons, not those of the non-Jews (Sichot Ha-Rav Tzvi Yehudah – Yom Ha-Atzmaut and Yom Yerushalayim, talk for Yom Yerushalayim p. 90 and Tal Chermon - Moadim, p. 218).
Our Rabbi also wrote that we should also be concerned about ripping our garments when not obligated and violating "Bal Tashchit" (wanton destruction of items) when the whole prohibition for tearing when seeing the Temple in a destroyed state is a Rabbinic prohibition.
For a further survey of sources, see R Ari Enkin's writeup.
Unfortunately, the vast majority of Jews who visit the Wall are not (as yet) religious. Most of them don't know the rule that one should tear his clothes and, if they did, most would most likely not abide by it, so as not to ruin their clothes for the sake of a rule they don't consider obligatory.
(Among religious Jews who visit, many also don't know the rule; others have visited recently enough that there is no obligation to tear clothes. See also Zvi's answer.)
 Citation needed, but I think it's pretty obvious from looking around.