Is it bad design to use arrays within a database?

Whether an array is atomic depends on what you're interested in. If you generally want the whole array then it's atomic. If you are more interested in the individual elements then it is being used as structure. A text field is basically a list of characters. However, we're usually interested in the whole string.

Now - from a practical viewpoint, many frameworks and ORMs don't automatically unpack PostgreSQL's array types. Also, if you want to port the database to e.g. MySQL then you'll

Likewise foreign-key constraints can't be added to an array (EDIT: this is still true as of 2021).

Short answer to the title: No

A bit longer answer:

You should learn to use arrays when appropriate. Arrays are not bad design themselves, they are as atomic as a character varying field (array of characters, no?) and they exists to make our lives easier and our databases faster and lighter. There are issues considering portability (most database systems don't support arrays, or do so in a different way than Postgres)

Example:

You have a blog with posts and tags, and each post may have 0 or more tags. The first thing that comes to mind is to make a different table with two columns postid and tagid and assign the tags in that table.

If we need to search through posts with tagid, then the extra table is necessary (with appropriate indexes of course).

But if we only want the tag information to be shown as the post's extra info, then we can easily add an integer array column in the table of posts and extract the information from there. This can still be done with the extra table, but using an array reduces the size of the database (no needed extra tables or extra rows) and simplifies the query by letting us execute our select queries with joining one less table and seems easier to understand by human eye (the last part is in the eye of the beholder, but I think I speak for a majority here). If our tags are preloaded, then not even one join is necessary.

The example may be poor but it's the first that came to mind.

Conclusion:

Arrays are not necessary. They can be harmful if you use them wrong. You can live without them and have a great, fast and optimized database. When you are considering portability (e.g. rewriting your system to work with other databses) then you must not use arrays.

If you are sure you'll stick with Postgres, then you can safely use arrays where you find appropriate. They exist for a reason and are neither bad design nor non-compliant. When you use them in the right places, they can help a little with simplicity of database structures and your code, as well as space and speed optimization. That is all.

Short answer: Yes, it is bad design. Using arrays will guarantee that your design is not 1NF, because to be 1NF there must be no repeating values. Proper design is unequivocal: make another table for the array's values and join when you need them all.

Arrays may be the right tool for the job in certain limited circumstances, but I would still try hard to avoid them. They're a feature of last resort.

The biggest problem with arrays is that they're a crutch. You know them already and you want to use them because they're familiar to you. But they do not work quite like you expect, and they will only allow you to postpone a true understanding of SQL and relational databases. You're much better off waiting until you're forced to use them than learning them and looking for opportunities to rely on them.