How unsafe is to share parts of a password The best way to ask this question and have a concrete answer is to set a very specific example. This is the example I would be interested in reading an answer to:

Consider that my friend Alice has several (~5) hard drives encrypted with Truecrypt. Alice is using AES (256-bit, 14 rounds) with RIPEMD-160. I could see that she has a script for mounting the different drives -I guess that because she's pretty lazy and just wants to type in one password. Of course, every disk is salted differently, etcetera.

From her script, I can figure out that the password for each drive comes from concatenating a common string (for all drives) with a fixed string set in the script (different for every drive). I don't know the length of the common string (it's user typed), but the fixed strings are like 6-10 characters long. 

Let's then first assume that I don't know the common part of the password, but I know the fixed part for every disk because I see it in the script.

So the question is, and don't be afraid of diving into explaining the mathematical reason- how insecure is the fact that she uses a common string, and how could I attack that flaw?

And secondly, what if I didn't know the fixed part for every hard drive but I knew that they begin with the same common part (which value is unknown too)?
If I know your password is k bits long and I know the m bits, then I have to brute force the rest of the k-m bits. That's the standard for any algorithm and independent of whether this is prefix, suffix etc. This means that I have $2^{k-m}$ tests to make. 

However, you can even do better, You can say that you have a decayed version of the key, or you have partial key exposure, you have some plaintexts and the according ciphertexts. In this scenario, there are several effective attacks, even for AES, with different approaches, even SAT solvers. Have a look at the following papers:


Halderman, J. Alex, et al. "Lest we remember: cold-boot attacks on encryption keys." Communications of the ACM 52.5 (2009): 91-98. 
Kamal, Abdel Alim, and Amr M. Youssef. "Applications of SAT solvers to AES key recovery from decayed key schedule images." Emerging Security Information Systems and Technologies (SECURWARE), 2010 Fourth International Conference on. IEEE, 2010.
Liao, Xiaojuan, et al. "Using MaxSAT to Correct Errors in AES Key Schedule Images." Tools with Artificial Intelligence (ICTAI), 2013 IEEE 25th International Conference on. IEEE, 2013.


Of course there are limitations as you can see from the papers, but depending on how much bits you want to crack, it might be easier to go this way than brute force them.