In principle it is possible to have a PKI (signature, encryption) applet on any JAVA Card enabled SIM card. The problem is that the issuing instance - the MNO in most cases - is in charge of loading these applets and installing keys. So for the moment there is no practical solution present in the field. There are other so called "secure elements" which are either embedded in the phone (same problem) or can be attached to it, like microSD including smart card chip or a "sleeve" attached to the phone also containing a secure element (i.e. smart card chip).

In theory, smartphones can be used like desktops to produce signatures --see Tim Bray's send/receive demos on Youtube.  That implies the device is kept in a trusted state and users understand enough of PKI.

Additional tokens carrying better secured chips and key pairs usually miss a display that can reliably convey to users the meaning of what they sign.  So they're bound to solve only part of the problem.

https://www.youtube.com/channel/UCZYI0JHoJCILEegRJsWufXQ

Basically there should not be a problem to deploy digital signatures in mobile devices including smartphones. 

I believe your question is ways broader that smartphones and digital signatures. The real issue is that of trust. Trust is non transferable and it cannot be inherited. For example if A trusts B and C trusts B it does not imply that C trusts A. Then, "no trust" is very clear while "trust" has an infinity of gray areas like who is trusted, for what purposes and what extent. This issue applies also in the regard of ID certificates the question being the trust in the certificate authority (CA) and then in the certificate from such CA as well as in digital credentials digitally signed.

Reliance on digital signatures for other than short lived transactions is problematic. There are legal issues as well as technology escrow issues.  For example, if a signature was valid 3 years ago, but the keys used are no longer valid or the technology used to generate the signature is deprecated, how can one convince a court of the veracity of the signature?  

On the technical side, most SIM cards contain only a symmetric key which is used as a shared secret with carriers to generate and respond to challenges as a mechanism to prevent cloning, so the existing keys and algorithms available on existing SIM cards are likely lacking for this purpose if you are really contemplating a cryptographic digital signature. More promising is the study of mobile payment systems currently used in Africa.  Clearly, it seems to be working, though flaws may exist.

Finally, if one looks at the history of certificate issuing authorities, one comes away somewhat depressed since it seems anyone can become an issuing authority and if your trusted certificate list contains any dubious issuing authorities, then that is a problem.

 The only way I have seen this used effectively is when one uses one's own certificate authority to sign the keys of trusted devices and then uses this as part of the connection requirement for the device via a mutually authenticated TLS or SSL connection.  This effectively limits all traffic from other than a device having an appropriately signed X.509 certificate and is an effective deterrent in this respect.  However, one would need to be able to generate and store the key material securely and would need to set up a strategy for generating and installing the certificates.  Symantec and others have cloud services that could facilitate this to scale, and would be worth a look.

I am assuming your intention is to allow a customer to use his mobile phone for digital signing during electronic banking transactions. 

In the above case, the main issues are: How to store the private key securely and safely ? and How to access the private key in a secure manner during the digital signing process ? 

World over, Cryptographic USB tokens (which are cryptographic co-processors having their own memory with USB interface) are popular, but in mobile we want to avoid carrying another device, like that. In such a case the following solutions are being explored.

However each of the above has its own pros and cons, including costs. While costs can be offset by high volumes in future, the convenience and ease of use will hold the key to successful adoption.