Is it true that the Prophet forbade to write hadiths?
No it is not true - in fact the Prophet (saws) taught his Companions (raa) these hadith formally in circles of knowledge. He spent a lot of time commanding them to convey his words, and he used to quiz the Companions also about what they had learned earlier. There was a group of people whose full-time job was to learn and record - commonly known as As-hab as-Suffah. Some of the famous Companions from this group include Abu Huraira, Jabir b. Abdullah, Mu'adh b. Jabal, Abdullah b. Mas'ud and others.
In their time, the primary way of recording something and propagating it was by memorizing it and oral transmission. Needing to write something down was seen as a weakness - these people had extraordinary memories. Secondly, another method they used to preserve the hadith was by practising them and making them a part of their lives. This is another effective way of preservation and propagation.
Initially, in the early stages of Islam, when the style and pattern of the Qur'an wasn't that clear yet, the writing down of hadith was prohibited so as to prevent them from being mixed with the words of the Qur'an. Later, the Prophet (saws) himself commanded Amr b. al-Aas to write down the hadith he was learning. Other instances of explicit permission and commands to write hadith include the hadith in Jami' at-Tirmidhi about seeking help from "the right hand" - i.e. to preserve knowledge by writing it down. The Prophet (saws) explicitly told Rafi' b. Khadij that he could write down the hadith he was learning and there was no harm in it. Anas (ra) narrated that the Prophet (saws) told them to preserve knowledge by writing it down. And as mentioned earlier, Amr. b. Aas was commanded to write hadith. He asked whether he should write down everything, whether the Prophet (saws) was content or upset, and the Prophet (saws) replied by saying yes, everything, nothing comes out of my lips except it is the truth.
The hadith were written down since the earliest times - they were only compiled later in the times of the Companions and later, the Imams. There is a hadith from Anas (ra) that says that around 60 men would gather to review the ahadith of the Prophet (saws). 'Umar and Abu Musa al-Ash'ari used to review hadith until the early morning. For example, Abdullah b. Amr b. Aas had a compilation of hadith with over 5000 of them called As-Sahifa as-Sadiqa, which the Prophet (saws) himself reviewed. Anas b. Malik had a compilation of over 2000, Jabir b. Abdullah over 1500, Aisha (ra) over 2000, and ibn Abbas and ibn 'Umar over 1500.
The first person to start an official compilation of hadith was Umar b. Abdul Aziz, just a few decades after the Prophet (saws) when he appointed the well-known muhaddith Ibn Shihab Az-Zuhri to oversee this project.
The fact is that the hadith were memorized systematically by the Companions and passed down just as the Qur'an was memorized and passed down.
The main source for this answer is notes from a class on hadith studies I took a few years ago. The textual sources for that class were mainly the book by Dr. Mustafa al-Azami on sciences of hadith and Shah Waliullah's book on the history of hadith.
To be completely objective and neutral there is no clear and definite answer to this and here is why:
Abu Sa’id Khudri reported that Allah’s Messenger (may peace be upon him) said: Do not take down anything from me, and he who took down anything from me except the Qur’an, he should efface that AND narrate from me, for there is no harm in it and he who attributed any falsehood to me−and Hammam said: I think he also said:” deliberately” −he should in fact find his abode in the Hell−Fire. (Source: Sahih Muslim Book 42, Number 7147)
Narrated by Zaid ibn Thabit Al-Muttalib ibn Abdullah ibn Hantab said: Zaid ibn Thabit entered upon Mu’awiyah and asked him about a tradition. He ordered a man to write it. Zaid said: The Apostle of Allah (PBUH) ordered us not to write any of his traditions. So he erased it. (Source: Sunan of Abu-Dawud Hadith 3640)
But on the other hand:
I swear by the One in whose hands is the soul of Muhammad: nothing comes out from these two (lips) except truth. So, do write. [Sunan Abu Dâwud; Tabaqât ibn Sa'd; Mustadrik-ul-Hâkim]
Narrated by Warrad Muawiya wrote to Mughira, “Write to me what you heard the Prophet saying after his prayer.” So Al-Mughira dictated to me and said, “I heard the Prophet saying after the prayer, ‘None has the right to be worshipped but Allah Alone Who has no partner. O Allah! No one can withhold what you give, and none can give what you withhold, and the fortune of a man of means is useless before you (i.e., only good deeds are of value). “(Source: Sahih Bukhari Hadith Volume 8. Number: 612)
Some scholars tried to explain that it was forbidden to write down the hadiths while the Qu'ran wasn't finished, but I find that excuse weak as there is no way someone would confuse the Qu'ran with the hadiths and that Allah protects the Qu'ran and the false Prophet Musaylimah tried and failed to alter the Qu'ran.
People tend to either reject all the hadiths or believe all of them, I think that we can have a middle ground. It's pretty obvious that the hadith collections are far from perfect, even if the Prophet (pbuh) allowed the hadiths the importance and the usage of the hadiths was made-up after his death (you can read about the canonization of the hadiths and the formation of the Islamic jurisprudence).
On top of that, despite the "great" methodology and "preservation" we are still changing the hadiths as sahih or daif to this day. I follow the hadiths as long as they don't contradict the Qu'ran, as hadiths can be wrong but the Qu'ran cannot. I find it very strange how some scholars are denying any hadith criticism, aren't they afraid to spread lies about the Prophet(pbuh)? To quote Imam Malik:
"The shield of the scholar is, 'I do not know,' so if he neglects it, his statement is attacked."
And Allah knows best.
The Prophet ﷺ forbade writing hadiths earlier on, then he allowed it after. having said that, the main way of passing the traditions of the Prophet ﷺ on to us was verbal. The Prophet ﷺ used to speak slowly and clearly to the extent that 'Ā'isha said that he "used to utter so distinctly that if one intended to count (the words uttered) he would be able to do so" (see Sahih Muslim 55/91), and Anas ibn Mālik said he used to repeat what he said thrice (see Jami' at-Tirmidhi 1/46/3640). This was aimed at allowing those who were authorized to write to do so, and the rest could memorize what he said.
In the first years of revelation, the literate documented the hadiths. The Prophet ﷺ saw that some were mixing the Qur'an (revealed by words) and the hadiths (revealed by meaning only, but the words are of the Prophet's), so he forbade writing hadiths:
عن أبي سعيد الخدري أن رسول الله صلى الله عليه وسلم قال: لاَ تَكْتُبُوا عَنِّي وَمَنْ كَتَبَ عَنِّي غَيْرَ الْقُرْآنِ فَلْيَمْحُهُ وَحَدِّثُوا عَنِّي وَلاَ حَرَجَ وَمَنْ كَذَبَ عَلَىَّ - قال همام أحسبه قال - مُتَعَمِّدًا فَلْيَتَبَوَّأْ مَقْعَدَهُ مِنَ النَّارِ
Abu Sa'id Khudri reported that Allah's Messenger (ﷺ) said: Do not take down anything from me, and he who took down anything from me except the Qur'an, he should efface that and narrate from me, for there is no harm in it, and he who attributed any falsehood to me — and Hammam said: I think he also said:" deliberately" — he should, in fact, find his abode in the Hell-Fire.
While the Qur'an continued to be documented on paper, leather, wood, etc., the Islamic scholarship based on the hadiths turned predominantly oral, save for a few exceptions. Later, when it became easy for the companions to decipher the Qur'an from the hadith, the Prophet ﷺ removed the restriction to write:
عن عبد الله بن عمرو قال: كنت أكتب كل شيء أسمعه من رسول الله صلى الله عليه وسلم أريد حفظه فنهتني قريش وقالوا: أتكتب كل شيء تسمعه ورسول الله صلى الله عليه وسلم بشر يتكلم في الغضب والرضا، فأمسكت عن الكتاب، فذكرت ذلك لرسول الله صلى الله عليه وسلم فأومأ بأصبعه إلى فيه فقال: اكْتُبْ فَوَالَّذِي نَفْسِي بِيَدِهِ مَا يَخْرُجُ مِنْهُ إِلاَّ حَقٌّ
Narrated Abdullah ibn Amr ibn al-'As: I used to write everything which I heard from the Messenger of Allah (ﷺ). I intended (by it) to memorise it. The Quraysh prohibited me saying: Do you write everything that you hear from him while the Messenger of Allah (ﷺ) is a human being: he speaks in anger and pleasure? So I stopped writing, and mentioned it to the Messenger of Allah (ﷺ). He signalled with his finger to him mouth and said: Write, by Him in Whose hand my soul lies, only right comes out from it.
There were specific companions that documented lots of hadiths in writing, e.g., 'Abdullah ibn 'Abbās, 'Abdullah ibn 'Amr, 'Abdullah ibn Mas'ūd, Abu Hurairah, 'Amr ibn al-'Ās, Anas ibn Mālik, Ibn Jandab, Jābir ibn 'Abdullah, and Sa'd ibn 'Ubāda al-Ansāri, among others.
The study of the sanad (chain of narration) and its authentication started during the era of 'Uthmān ibn 'Affān, about 20 years after the death of the Prophet ﷺ, when the need to judge the degree of credibility of a hadith rose as a result of the sectarian split over issues. This laid the foundation of the science of textual criticism of hadith. There was, however, no systematized compilations of the hadiths at the time. The compilation of hadiths as ordered by 'Umar ibn 'Abdul-'Azīz and carried out by Ibn Shihāb az-Zuhri.
It was not until Ar-Rabī' ibn Subaih (Arabic: الربيع بن صبيح) and Sa'īd ibn Abu 'Urūba (Arabic: سعيد بن أبي عروبة), among others, started to systemize and classify compilations by a taxonomy of topics that they adopted. This helped the following generation (e.g., Imam Malik ibn Anas in his Muwatta') to authenticate hadiths and derive rulings based on their classified works. Soon after, compilations were classified in accordance with the need to establish a system jurisprudence teachings (e.g., Malik, Al-Awzā'i, Sufyān ath-Thawri, Ibn Hanbal, Ibn Dinār, Ibn Abu Shaiba, etc.).
Based on the work documented over the preceding century, emerged a number of scholars that applied the conditions for authentication of hadiths to the compilations available at the time, e.g., Mohammad ibn Isma'īl al-Bukhāri, Muslim ibn al-Hajjāj, Abu 'Abdur-Rahmān Ahmad ibn Shu'ayb ibn 'Ali ibn Sīnān ibn Bahr An-Nasā'i, Abu Dawūd Sulaymān ibn al-Ash'ath al-Azdi as-Sijistani, Abu 'Īssa Mohammad ibn 'Īssa at-Tirmidhi, and Abū 'Abdullah Mohammad ibn Yazīd ibn Mājah, among others. From this group, two scholars saw the need to have a tighter application of the methodology already in place to include in a book that what was in it was definitely authentic, albeit that not all authentic was in it. The two scholars, Al-Bukhāri and Muslim, embarked on writing their books more or less at the same time, which later were established as the most authentic Islamic literature after the Qur'an (Sahih Al-Bukhari and Sahih Muslim).
The fabricated hadiths started during the time of 'Uthman by people simply lying and saying that the Prophet ﷺ said so and so to prove their sectarian views. The misinterpretations are more or less the same thing: relying on non-qualified people to explain them. We have books on hadiths interpretation that are reliable, e.g., Fat'h Al-Bari (for Sahih al-Bukhari), and Sharh an-Nawawi (for Sahih Muslim), etc. These are the books that we should rely on for interpretation.