Can I write, "Classes will be held Monday, Friday and Saturdays." when I am referring to a recurring weekly event for each day?
Since this is a weekly event, you have the option of either saying:
Classes will be held Monday, Friday and Saturday
or all plural:
Classes will be held Mondays, Fridays and Saturdays
You will find both usages are common in spoken and written English. In both business writing and conversation, it is very common to drop the plural and use all singular.
Here are some examples:
Classes taught on Tuesday–Thursday should be taught for 75 or 80 minutes beginning at 8:00 a.m.
--- Office of the Registrar, University of Illinois
Most classes meet Monday, Wednesday and Friday. Other classes meet Tuesday and Thursday.
Q: What do the abbreviations mean on the class schedule? What is TR?
A: TR means the class meets on Tuesday and Thursday every week. MWF means the class meets every Monday, Wednesday, and Friday.
UNCA’s class schedule uses R as the abbreviation for Thursday. If a class is listed as meeting TR, it meets on Tuesday and Thursday.
--- Schedule Facts that Freshmen (sic) Should Know (pdf), University of North Carolina, Asheville
For the most part, classes will either meet on Monday, Wednesday, and Friday for 50 minutes each day, or meet on Tuesday and Thursday for 1 hour and 15 minutes each day.
And some nice counterexamples:
This semester from 8-11 p.m. on Tuesdays and Thursdays, students can use the basketball courts in the Melo Center for pick-up basketball.
That is a special event, not a regular occurrence.
The Admissions Office presents Transfer Tuesdays & Thursdays [...] On select Tuesdays & Thursdays an Admissions Counselor will introduce you to Rowan, share a presentation, and offer you a campus tour in settings designed specifically for transfer students.
--- Rowan University, New Jersey
Notice this happens on select(Definition) Tuesdays and Thursdays, so this is not an every week event -- use plural.
Metabolic Conditioning Tuesdays & Thursdays at the SDC.
- Join Heather for Metabolic Conditioning Tuesday & Thursday mornings at 6 a.m. this summer in the SDC Studio!
--- Michigan Technological University, Michigan
This example uses both -- Plural in the headline, then singular in the text. Singular is used because "Tuesday and Thursday" modify "mornings", which is the noun.
Notice the parallelism. Singular and plural remain constant in the same list -- they are not mixed in the same phrase.
Wait, what about the punctuation?
Some of the examples used dashes ("Tuesday–Thursday"), some used commas ("Monday, Wednesday and Friday") and some used slashes ("Tuesday/Thursday"). Does this matter?
Some readers will consider it the most correct to use commas, because it's a conventional list (Alice, Bob, Charlie, and Delia) but you'll often see people using dashes or slashes to separate days of the week, as well as other technical items (valid Tuesday/Thursday, with 1–4 gallons of water, using versions 1.0/2.0/3.5 of the policy.)
Stick to commas and singular and you're fine.
Classes will be held Monday, Friday and Saturday.
You need 'Mondays, Fridays and Saturdays' for the meaning you want. What you had could be interpreted as the next Monday, next Friday, and then Saturdays on a continuing basis.
As a general rule, try to use identical parts of speech, verb tenses, plural nouns, etc. when showing some common feature in a series of items. It helps readers notice that some commonality exists and results in writing which is easier to understand.