The purpose of bedikat ĥametz is to find pieces of ĥametz that are a kezayit or larger, for one only violates bal yera’eh and bal yimatzei if there is a kezayit or more of ĥametz in one place in his home. Therefore, any part of the house into which people sometimes bring ĥametz must be searched. In a house where children are present, one must search all of the places that children enter; however, one need not search closets or shelves that are out of the reach of children.
Some poskim take the stringent position that the purpose of the search is to ensure that not a single edible crumb of ĥametz remains in one’s possession, for if even one crumb remains, somebody might mistakenly eat it during Pesaĥ and thus transgress a Torah prohibition. Even though there is no punishment for eating less than a kezayit of ĥametz, the Torah nonetheless prohibits it. According to this position, one must search the entire house meticulously and keep an eye out for even small crumbs that might be ĥametz. Such a search in a normal house should take at least two hours. However, even according to stringent opinions, one need not search for crumbs so small that they are not recognizable as food. Likewise, there is no need to search for crumbs so filthy that they are inedible. For example, it is not necessary to inspect the cracks between floor tiles, because the crumbs there are repulsive and not fit to be eaten.
In practice, one who chooses to be stringent and search for crumbs smaller than a kezayit is praiseworthy, but the halakha follows the lenient opinion. The reason is that the obligation to search for ĥametz is rabbinically ordained. According to the Torah, one who mentally nullifies his ĥametz has already avoided the prohibition of ĥametz and need not search his home. It is the Sages who ordained that, in addition to the bitul, we must seek and destroy ĥametz. Whenever there is a disagreement about a rabbinic enactment, the lenient opinion is generally preferred.
All this applies to the house in general, but it goes without saying that any object that will come into contact with food during Pesaĥ must be thoroughly cleaned, for even the slightest amount of ĥametz renders food forbidden on Pesaĥ. Therefore, the dining table, countertops, and cabinets must all be cleaned so well that not a single crumb of ĥametz remains.
. SAH explains in 446, Kuntrus Aĥaron 1, that according to most Rishonim, there is no need to burn a piece of ĥametz smaller than a kezayit (see Birur Halakha 45:1 on the topic of dough in the cracks of a kneading tub). MB 442:33 cites a dispute about whether one must destroy a piece of ĥametz smaller than a kezayit: some say that since one transgresses a prohibition by eating it on Pesaĥ, he must burn it beforehand; others are more lenient and would not require him to burn it.
SHT ad loc. mentions that the practice is to be stringent and burn this ĥametz, but does not discuss whether one must search for crumbs smaller than a kezayit during bedikat ĥametz. However, it seems from the comments in 442:60 that the main point of the bedikat ĥametz is specifically to find ĥametz that is larger than a kezayit, and indeed this is the opinion of Pri Ĥadash and many other halakhic authorities. On the other hand, Ĥayei Adam (109:6) rules stringently that one must search even for small crumbs. This is also the view of Ĥazon Ish (OĤ 116:13, 17).
Ostensibly, it is possible to connect this dispute to a disagreement about the purpose of bedikat ĥametz. According to Rashi, the purpose of bedikat ĥametz is to prevent one from transgressing the Torah prohibitions of bal yeira’eh and bal yimatzei, which, according to almost all authorities, one does not violate with less than a kezayit of ĥametz. Sha’agat Aryeh explains that small amounts of ĥametz do not combine (“lo ĥazi le-itztarufei”) to cause one to transgress bal yera’eh and bal yimatzei, unlike small amounts of forbidden foods, which combine to the size of a kezayit (according to Ĥakham Zvi, since there is no action involved in the violation of bal yera’eh and bal yimatzei, the principle of “aĥshevei” does not apply).
On the other hand, according to Tosafot, since the purpose of bedikat ĥametz is to prevent one from eating any ĥametz that remains in his house on Pesaĥ, one would thus be required to check for even small crumbs of edible ĥametz (and remove them from his home). Since we follow both Rashi’s and Tosafot’s reasons, as Ran records both, perhaps one must be stringent in this matter. However, it seems that even according to Tosafot, despite the fact that one would violate a Torah prohibition by eating less than a kezayit of ĥametz, the Sages did not require one to check for such crumbs, since eating a piece of ĥametz this size does not incur a penalty of karet. Additionally, Tosafot’s concern about one finding ĥametz in his house on Pesaĥ and eating it applies mainly to a nice, substantial piece of ĥametz; regarding a small crumb of ĥametz, the concern is not as severe. Even if one were to find, say, a small crouton somewhere in his house, it is not likely that he will eat it, since it was found where food is not normally stored. Furthermore, even if he does eat this small crouton, he arguably did not intend to eat it; rather, he was cleaning, and instead of throwing the crouton in the garbage, he put it in his mouth. Such an action is not a Torah prohibition, and not something for which the Sages would mandate bedikat ĥametz. Nevertheless, perhaps one who normally eats these types of crumbs should be stringent in accordance with the ruling of Ĥazon Ish.
See also Hilkhot Ĥag Be-ĥag ch. 6 n. 2, which discusses the two opinions in this matter and proves the lenient approach from the implication of Pesaĥim 4a that bedikat ĥametz lasts for less than an hour.
See also Sidur Pesaĥ Ke-hilkhato §13 n. 39, which rules according to the stringent opinion that one must check for crumbs, and Bedikat Ĥametz U-vi’uro 2:1, which is very strict about crumbs. However, since this is a dispute about a rabbinic injunction, practice follows the lenient approach. Nevertheless, one must differentiate between places to which he brings ĥametz and the rest of the house, similar to the distinction made regarding koshering utensils and bedikat ĥametz.