The best mutual fund [MF] is Franklin Templeton, and the worst, Reliance purely from the point of view of settlement of claims arising out of death of the account holder, IMO.
My late father had investments in many plans and schemes across numerous mutual funds. After his death, my mother became the sole holder of these accounts. She expired in June, 2012. After verification of records, I came to know that I am the nominee for most of these accounts.
So, I started the claim process of these accounts from July 2012, which continues even today! I have the first hand experience of settlement of claims due to death by Franklin Templeton, UTI, SBI, HSBC, Sunlife Birla, Tata, Sundaram and Reliance MF. Multiply these by 4 or 5 to get an idea of total number of accounts in these MF!
Before going into the details, let me clarify that my parents expired and cremated in Chennai. Corporation of Chennai has made available online all the birth and death certificates in their website. These birth/death certificates can be downloaded by anybody from the Corporation of Chennai website, which contain unique ID. The computer-generated certificates clearly state that authenticity of the certificates could be verified from the Corporation of Chennai website by typing the unique ID in the required field. So, I downloaded the death certificates of my parents, obtained attestation of my signature by the bank manager in the required format, took photo-copies of the other required documents like PAN card, ration card, etc. and started the claim process in various MF.
I started with Franklin Templeton first. When I explained and submitted the required documents, I was asked, whether I would like to continue to hold on to the accounts after transmission into my name or redeem the balance amount; I told them that I prefer the latter option. Then, I was told that KYC norms were not required and they would take about 10 working days to process the claim. True to their words, I got the proceeds credited to my bank account within 10 days.

I thought the same thing would happen in other MF; but, I was totally wrong.
All other MF insisted on transmission of these accounts in my name, and later, I could redeem them if I wish for that. Okay, no issue with that. But, others insisted on my KYC norms for MF. When I told them I already have my demat account with Kotak Securities registered with NSDL, I was told that won't do; I must register separately for MF. Then, I submitted KYC application with supporting documents to CAMS, who are one of the registration authority for KYC for MF, I suppose. My details got verified and I received an acknowledgement letter from CAMS. Enclosing this acknowledgement letter, I started claiming in other MF. One-by-one, I got them transferred in my name, and later, I redeemed them again, one-by-one.
Now, come to the peculiar case of Tata MF. There were 4 accounts in different schemes/plans. Three were dealt by their registrars, CAMS, where I got the work done without any issue; in case of TATA Contra Fund, which was dealt by Karvy, I was troubled to the maximum extent. After submission of the documents, I received a letter asking me to submit KYC acknowledgement letter; I submitted KYC acknowledgement letter issued by CAMS. After a month or so, again I was asked to submit KYC acknowledgement letter from CVRL! When, I personally visited TATA MF office at Kodambakkam High Road, Chennai, the customer care staff called the concerned in Karvy and asked to process my claim as CAMS letter was more than sufficient; CVRL registration was not required. Then, there was absolute silence from their end. After a month, when I visited TATA MF office, I was told the account was already transferred in my name and I was given a statement, which I filled in then and there only for redemption.
Now, the worst of all. I visited the investor centre of Reliance MF at T Nagar, Chennai with all the documents. The customer care staff examined all in detail; he personally typed the unique ID of the death certificates of my parents in the Corporation of Chennai website and took prints and returned the original death certificates submitted by me. He also issued an acknowledgement letter for all the accounts. After a month, I receive five letters from the registrar [who else but Karvy!] asking me to submit death certificates attested by a gazetted officer or notary public as I submitted photocopies! I immediately sent emails to the customer care of Reliance MF and Karvy explaining that they were the prints taken by their investor centre after verifying the authenticity in the Corporation of Chennai website and since they were computer-generated ones no signatures were required. Again silence from Karvy/Reliance MF for another month. Again, I received the same letters from Karvy asking me submit attested copies of death certificates as I submitted photocopies. I got fed up, took attestation from a known gazetted officer, submitted them, got the accounts transferred in my name and applied for redemption a few days back. I am yet to get the amount credited to my bank account. Reliance MF/Karvy are the worst as far as claims settlement is concerned.