At the moment, there are a lot of services that have the opportunity to store cryptocurrency. In fact, the choice of storage should be taken seriously and it is necessary to learn as much information about cryptocurrencies as possible beforehand.

If this question is taken seriously, there are two aspects: where you can store cryptocurrency and where you should store it.
All cryptocurrency storage services are divided into two classes: custodial and non-custodial. Some sources also divide them into "hot" and "cold", this is not an accurate comparison, but said in simpler words. In fact, "hot" wallets can also be non-custodial - the pattern is described below.
"Hot" and "cold" wallets differ from each other in terms of connection mode - the former is always online, while the latter uses a connection to the blockchain only when sending or receiving data.
The differences between custodial and non-custodial are not so noticeable at first glance, but if you apply them, the difference is noticeable. Custodial services are analogous to the banks that exist in normal life. This way your cash is deposited into the custody of a certain organization (service) and you can dispose of your tokens. A non-custodial service is a special interface that gives you access to a wallet in the official blockchain of any cryptocurrency. In this case, the service has access to the wallet management for a certain time, using your private key (or seed phrase).

Both options have certain pros and cons

Custodial services are much easier to manage. If you lost your password, you can restore the access, it requires proof of identity. Basically, these services are a whole ecosystem with different financial instruments and you can buy cryptocurrency with your bank card. But their biggest disadvantage is centralization - which means that there is a certain organization that has the right to manage the funds on your behalf.
Many crypto-enthusiasts express this phrase: "not your keys, not your coins. Due to recent events in the world, this topic is very relevant, as many accounts have been frozen due to the bias of certain entities.
Non-centralized services are more on the side of the original idea of cryptocurrencies - they are for economic independence and no transaction censorship. If even that is not as easy and convenient as custodial. In the absence of the seed phrase, access to the wallet will be completely lost, but still many experts recommend using these. It takes a bit of work and doing everything not to lose the secret phrase, but in the end it will be safe.The article uses the word "services", not "wallets", because cryptocurrency can be stored: on an exchange, in a wallet, in various payment systems or other services. Let's look at each of the categories below:

Exchanges

Cryptocurrency exchanges can be centralized (CEX) and decentralized (DEX). The first includes the popular Binance, Coinbase and others. On such exchanges it is easy to work with cryptocurrencies, as well as easy to input and withdraw fiat currencies. The most important disadvantage of a centralized exchange is the custodial service.
It is not recommended to store assets in decentralized exchanges. If you look on the other side, it's not even an exchange, it's a software package of automated algorithms. These algorithms provide for the automatic exchange of various tokens between each other. Therefore, despite the calls of crypto-enthusiasts to switch to DEX, it is impossible to do so.
The functionality of such services is limited to the ability to exchange tokens, and only those tokens that are in blockchains that support smart contracts. The problem is solved by tokenization of cryptocurrencies and because of this there are difficulties in working with little-known tokens, as well as no possibility to work with fiat currencies.
For example, Wrapped Bitcoin token (WBTC) was created for Ethereum ecosystem, its value is secured by the value of frozen bitcoins existing for that purpose.

Wallets

Cryptocurrency wallets can be categorized in many ways, in addition to custodial/non-custodial and hot/cold, they can also be categorized within each category.
The safest wallet is "cold", but such protection is redundant and inconvenient. The optimal wallet is considered a non-custodial wallet with the possibility of permanent connection, i.e. from the category of "hot".
Wallets vary in the way they are used: browser-based (MetaMask, Brave Wallet, Coinbase Wallet), desktop-based (Bitcoin Core, Avax Wallet, Electrum, Jaxx), online (MyEtherWallet, blockchain.com), hardware-based (Ledger, Safepal, Trezor, Keepkey) and mobile (Trust Wallet , Jaxx, Electrum, Phantom Wallet). These are only examples of the most popular at the moment.
Multicurrency cryptocurrency wallets can be considered a separate category of cryptocurrency wallets. Despite their convenience, you have to be careful, because they can be both custodial and non-custodial. If you can't export the secret keys from the wallets created in the app when you use them, it will be a custodial service.

Payment systems

Cryptocurrency can be stored on the balance of electronic payment systems (PayPal, WebMoney, Advanced Cash, Payeer, Perfect Money, etc.).
Such services are simple and convenient, with low commissions and work with fiat currency, but rather insecure due to centralization.

Third-party services

There is also an opportunity to store cryptocurrency at Western brokers or in the accounts of progressive banks. Examples of such services will be considered below.