What is the biblical basis for only keeping the weekly Sabbath while disregarding all the other feasts/holy convocations (Lev 23, Num 28, 29)? Leviticus 23 introduces several appointed festivals/holy convocations. The chapter starts with:

The Lord said to Moses,
2 “Speak to the Israelites and say to them: ‘These are my appointed festivals, the appointed festivals of the Lord, which you are to proclaim as sacred assemblies. - Lev 23:1-2 (NIV)

Then the festivals / holy convocations are enumerated:

The weekly Sabbath (Lev 23:3)
The Passover and the Festival of Unleavened Bread (Lev 23:4-8)
Offering the Firstfruits (Lev 23:9-14)
The Festival of Weeks (Lev 23:15-22)
The Festival of Trumpets (Lev 23:23-25)
The Day of Atonement (Lev 23:26-32)
The Festival of Tabernacles (Lev 23:33-44)

Similarly, in Numbers 28 and 29 we find a similar list:

Daily Offerings (Num 28:1-8)
Sabbath Offerings (Num 28:9-10)
Monthly Offerings (a.k.a. New Moons) (Num 28:11-15)
The Passover (Num 28:16-25)
The Festival of Weeks (Num 28:26-31)
The Festival of Trumpets (Num 29:1-6)
The Day of Atonement (Num 29:7-11)
The Festival of Tabernacles (Num 29:12-40)

It's not rare to come across Christians who claim that the weekly sabbath (Lev 23:3, Num 28:9-10) is still in effect, while all the other festivals and holy convocations are not (e.g. 1, 2, 3, 4). What is the biblical basis for that claim?

As requested in the comments, I'll briefly summarize the core ideas presented by some of these sites:

This site (and this follow-up publication on the same site) basically claims that the weekly sabbath belongs to the moral law, whereas the remaining feasts and holy convocations belong to the ceremonial law, and therefore they conclude that the weekly sabbath still stands but the others are abolished because they were &quot;shadows&quot;.

This site essentially says the same thing: it talks about the concepts of &quot;ceremonial law&quot; and &quot;moral law&quot;, claims that the weekly sabbath in particular is &quot;moral&quot; rather than &quot;ceremonial&quot;, and, thus, concludes that the weekly sabbath is still in effect.
A biblical basis for some Christians to claim that the weekly Sabbath is still in effect while the Feasts of Leviticus 23 need not be kept is that it's not clear that the Sabbath is included in the feasts of Leviticus 23 as a feast.
It could be that verse 2 ends by indicating that what is to follow is &quot;concerning the feasts&quot;, verse 3 is a parenthetic instructing that Sabbath observance is not to be disregarded during the feasts, verse 4 begins with the introduction &quot;these are the feasts of the Lord&quot; and then verse 5 begins the instructions concerning the feasts.

23:2 “Speak to the people of Israel and say to them, These are the appointed feasts of the LORD that you shall proclaim as holy convocations; they are my appointed feasts. 
Lev 23:3  “Six days shall work be done, but on the seventh day is a
Sabbath of solemn rest, a holy convocation. You shall do no work. It
is a Sabbath to the LORD in all your dwelling places.
Lev 23:4  “These are the appointed feasts of the LORD, the holy
convocations, which you shall proclaim at the time appointed for
them.

Additionally, the question lists the feasts of Passover and of Unleavened Bread concurrently but if they are enumerated separately (as many hold they should be) then the feasts number 7 without the inclusion of Sabbath.  The two are often delineated in Leviticus 23:5-8 under the same (man-made) heading &quot;The Passover&quot; but they are clearly not concurrent.  The Passover is a single meal commemorating a single event and occurs on 14 Nissan and  Unleavened Bread is a seven day affair beginning on 15 Nissan:

In the first month, on the fourteenth day of the month at twilight, is the LORD's Passover.  And on the fifteenth day of the same month is the Feast of Unleavened Bread to the LORD; for seven days you shall eat unleavened bread. On the first day you shall have a holy convocation; you shall not do any ordinary work. But you shall present a food offering to the LORD for seven days. On the seventh day is a holy convocation; you shall not do any ordinary work.” - Leviticus 23:5-8

Unleavened Bread is an example of a feast that lasts multiple days, starting and ending on holy convocations.  One of the feast days is, by necessity, a weekly Sabbath.  It is not the only feast which also incorporates a weekly Sabbath and so lends credence to the idea of verse 3 being a parenthetic reminder to keep the weekly Sabbath during the festivals.
This explanation does nothing to instruct or suggest whether the Church need or need not keep any or all of them beyond that the audience intended in verse 2 is &quot;the children of Israel&quot; (which thing is an entirely different question).