Returning lost money that one found:[1]

Finds scattered bills or coins in public area: One who finds money [such as bills or coins] scattered [in a public area], and it is not inside of a purse/wallet, may keep the money for himself. This applies [irrelevant of the amount of money found[2] and applies] even if the money was found in a majority Jewish area.[3]

If the money contains a Siman/symbol of ownership:[4] The finder may keep the money for himself even if the cash contains a sign of ownership, such as the name of an individual.[5] [Certainly the number/value on a coin or bill is not considered a Siman to require one to return it to one who can testify to its value.] However, if the coin or bill contains a unique sign that is uncommon to be found, such as a cracked coin, then if found in a majority Jewish area, one is obligated to announce his finding and try and return it to the owner.[6] [Likewise, if a bill contains a unique writing that can be recognized by its owner, and is uncommon to be done to other bills, then it must be announced and returned. Thus, a Rebbe dollar that has a name and/or date written on it must be returned to its rightful owner]

If one finds money in a purse/wallet:[7] If one finds in a majority Jewish area a pouch, purse, or wallet that contains money, then, one is obligated to announce his finding and try to return it to the owner.[8] This applies even if one found the money resting outside of the wallet, but next to it, in a way that is evident that it fell from that wallet. The money is to be returned to whoever testifies towards a symbol found on the wallet/purse.

If one finds a pile of cash:[9] One who finds a pile of cash which was evidently consciously placed there temporarily and later forgotten, is obligated to announce his finding and try to return it to the owner, being that the money contains a symbol, such as the amount or location. One is to return the money to the person who gives the symbol of the amount of money found, or the location.

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[1] Admur Hilchos Metzia Upikadon 8; Michaber C.M. 262:6 and 11; Mishneh Bava Metzia 21a

[2] See Sefer Hashavas Aveida Kehilchaso [Friedlander]

[3] Admur ibid; Bava Metzia 24a

The reason: As people are accustomed to check their [pockets or wallets for their] money all the time, and thus, we can assume that the owner who lost the money already discovered that it was missing and gave up hopes of finding it, prior to it being found. [Admur ibid; Smeh 262:13; Tur 262; Rebbe Yitzchak in Bava Metzia 21b; Accordingly, by the time the money was found it was already Hefker, and the finder may keep it just as he may keep anything from Hefker.] The reason we assume that the owner who discovers his money is missing will automatically give up hope of finding it is because money does not carry a unique symbol of ownership which one can testify towards, as many coins look the same. [Admur ibid; Smeh 262:28; Ramban 25b; Braisa Bava Metzia 25b] There is also no way for the owner to give a symbol of where his money was found, as when money is found scattered, it obviously fell from him without him realizing it and he thus does not know the exact area of where it fell from him. Likewise, he does not rely on a symbol of the amount of money that was lost to not give up hope on having the money returned to him, as perhaps some of it fell in one place and more in another place. [Admur ibid; Michaber 262:3 and 9; Smeh 262:19]

[4] Admur ibid; Michaber 262:13; Braisa Bava Metzia 25b

[5] The reason: As we suspect that perhaps the owner wrote his name on a number of bills/coins, and the one that was found is not the one that he lost, but rather a different one which he wrote his name on and was already spent, and really fell from the person who he gave it to, as money is customarily spent by the owner. [Admur ibid; Michaber ibid; Smeh 262:28; Ramban Bava Metzia ibid; Nimukei Yosef Bava Metzia ibid] Accordingly, the coin/bill is still considered to be without a sign even if one’s name is written on it. [Ramban ibid] See Teshuvah of Igros Moshe printed in Hashavas Aveida Kehilchaso p. 54; Toras Haveida p. 9]

[6] Admur ibid; Rama 262:13; Ramban ibid; Nimukei Yosef ibid

The reason: As although all coins are meant to be spent [and hence there is room to suspect that the person who is claiming a sign of ownership really gave it to someone else as payment and it fell from their hands] nevertheless we do not suspect for this in this case [as we do not assume that people who claim to have lost an item, and give a correct symbol to be liars]. Now, this type of symbol [a crack in a coin] is uncommon to occur and thus there is no room at all to suspect [as we did regarding a name] that perhaps another coin of his also cracked [and that is the coin he lost, and not this one which fell from the person he gave it to]. [Admur ibid]

[7] Admur ibid; Michaber 262:20; Rambam 16:3; Mishneh Bava Metzia 24b-25a

[8] The reason: As a wallet or purse is an item that contains a symbol of ownership. [Admur ibid]

[9] Admur ibid; Michaber 262:11; Smeh 262:24; Mishneh Bava Metzia 24b