What should the Muslim do with regard to scholarly differences of opinion as to whether a hadeeth is sound or weak?

Firstly: 

There is no difference among the scholars between scholarly
differences of opinion as to whether a hadeeth is saheeh or da’eef, and
their differences concerning matters of fiqh. That is because the
classification of hadeeth as saheeh or da’eef is subject to ijtihaad and the
scholars vary in their knowledge of narrators and isnaads of hadeeth. What
one of them knows about the circumstances of a narrator may be unknown to
others, and what another finds of corroborating reports may not be available
to another. So their rulings on a particular hadeeth may differ for these
reasons. Sometimes all of them found the biography of a narrator and the
isnaads of a hadeeth, but they varied in the rulings as to whether it was
saheeh or da’eef according to their own ijtihaad with regard to evaluating
the narrator and according to their view as to whether the hadeeth is free
of any problems.  

Imam al-Tirmidhi said: 

The imams differed in classing men (narrators) as da’eef
(weak), as they differed in other aspects of knowledge.

Sunan al-Tirmidhi (5/756),
Kitaab al-‘Ilal at the end of al-Sunan. 

Explaining the reasons for differences of opinion among the
scholars, Shaykh al-Islam Ibn Taymiyah said: 

The third reason: Belief that a hadeeth is weak on the basis
of ijtihaad, but others disagreed with him, regardless of other isnaads, and
regardless of whether he or someone else was correct, or both of them were
correct according to those who said that every mujtahid is correct. That is
for several reasons, including: the one who narrated the hadeeth thought
that one of them was da’eef and the other thought that he was trustworthy.
Knowledge of narrators is a vast science, and the one who is correct may be
the one who thought it was da’eef, because he had come across a reason to
make him think so, or the other one may have been correct because he knew
that this reason may not be enough to regard him as da’eef, either because
that reason in and of itself is not a problem or because he had an excuse
which meant that he could not be classified as da’eef.  

This is a broad topic for scholars who study narrators and
their circumstances, and there may be consensus and differences just as
there are among other scholars in their own fields. 

(Another reason is) that he did not believe that the
muhaddith heard the hadeeth from the one from whom he narrated it, but
someone else believed that he did hear it for reasons which imply that. 

(Another reason is) that the muhaddith may have been through
two phases, one when he was sound and credible and another when there was
confusion about his narrations, such as he got mixed up and his books were
burned or lost, so what he narrated when he was sound and credible is saheeh
and what he narrated when he was confused is da’eef. So he (the scholar) may
not have known which of the two scenarios applied to this hadeeth, but
someone else knows that he narrated this particular hadeeth when he was
sound and credible. 

(Another reason is) that the muhaddith forgot that hadeeth
and did not remember it afterwards, or he denied having narrated it because
he thought that there was a problem which meant that he should leave this
hadeeth alone, but another scholar may think that it is something that is
valid and may be quoted as evidence. This issue is well known … and there
are other reasons too. 

Majmoo’ al-Fataawa
(20/240-242). 

Secondly: 

With regard to the Muslim’s attitude towards these
differences which occur among scholars with regard to whether a hadeeth is
saheeh of da’eef, it is the same attitude as that towards their differences
of opinion in fiqh. If he is qualified to distinguish between their
opinions, he may decide which of the two rulings concerning one hadeeth he
thinks is correct; if he is not qualified to do so, then he should follow
the opinion of a scholar (taqleed) and he should accept the verdict of the
one who he thinks is more religiously committed and has greater knowledge
concerning this matter. He should not be deceived by the fact that he is a
faqeeh or scholar of usool or mufassir, rather the one whose verdict of
saheeh or da’eef is followed should be prominent in the science of hadeeth,
and there is no sin on him if he follows a prominent scholar. If the hadeeth
is saheeh according to that scholar and he follows him in that, and it
contains a fiqhi ruling, then he must act upon it, but there is no sin on
him for not acting upon it if the hadeeth is da’eef. 

Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said: 

If the scholars differed concerning it in their fatwas or
what is heard in their exhortations and advice, for example, then he should
follow the one who he thinks is closer to the truth in his knowledge and
religious commitment. 

Liqa’ al-Baab il-Maftooh (no.
46, question no. 1136) 

See also the answer to question no.
22652 for a discussion of the
correct attitude towards scholarly differences . 

And Allaah knows best.