Is there any research of the similar capabilities for sha256?
Sort of. SHA-256 is not designed to deal with burst errors, so you cannot compare the burst error detection properties of a CRC to a hash like SHA-256. Such a comparison simply isn't possible. However, you can compare the error detection capabilities of CRC32 with a hash like SHA-256.
With an algorithm like CRC32, certain types of changes to the input are guaranteed to result in a different checksum value. Using a good polynomial (a value intrinsic to a CRC's ability to detect modifications) allows more common types of changes to be detected. With a cryptographic hash like SHA-256 (which attempts to model a random oracle) on the other hand, there is no such guarantee. Detection of changes to input is entirely probabilistic. A cryptographic hash is as likely to detect a change to a single bit as it is to detect the entire input being replaced with something entirely different.
The thing to remember is that, unlike a CRC where certain types of input are more or less likely to result in a collision (with certain types of input having a 0% chance of causing a collision), the actual probability of collisions for input to a cryptographic hash is a function of only the length of the hash.
So, what are these different types of algorithms used for? A CRC is good for efficient, fast detection of accidental changes (corruption, etc.), with single-bit changes being detected with 100% probability. A cryptographic hash is used when the input set is large enough that the birthday bound may be reached and a small hash size is likely to result in some collisions, or when an attacker might intentionally try to craft input to collide (for example when generating certificates or signing a document).