When should you use a one-to-one relationship with tables in your database?
You can implement necessary fields inside one table. Even if data becomes very large you can enumerate column names that you need in a SELECT statement instead of using SELECT *.
The "1 to 0..1" between super and sub-classes is used as a part of "all classes in separate tables" strategy for implementing inheritance.
A "1 to 0..1" can be represented in a single table with "0..1" portion covered by NULL-able fields. However, if the relationship is mostly "1 to 0" with only a few "1 to 1" rows, splitting-off the "0..1" portion into a separate table might save some storage (and cache performance) benefits. Some databases are thriftier at storing NULLs than others, so a "cut-off point" where this strategy becomes viable can vary considerably.
The real "1 to 1" vertically partitions the data, which may have implications for caching. Databases typically implement caches at the page level, not at the level of individual fields, so even if you select only a few fields from a row, typically the whole page that row belongs to will be cached. If a row is very wide and the selected fields relatively narrow, you'll end-up caching a lot of information you don't actually need. In a situation like that, it may be useful to vertically partition the data, so only the narrower, more frequently used portion or rows gets cached, so more of them can fit into the cache, making the cache effectively "larger".
Another use of vertical partitioning is to change the locking behavior: databases typically cannot lock at the level of individual fields, only the whole rows. By splitting the row, you are allowing a lock to take place on only one of its halfs.
Triggers are also typically table-specific. While you can theoretically have just one table and have the trigger ignore the "wrong half" of the row, some databases may impose additional limits on what a trigger can and cannot do that could make this impractical. For example, Oracle doesn't let you modify the mutating table - by having separate tables, only one of them may be mutating so you can still modify the other one from your trigger.
Separate tables may allow more granular security.
These considerations are irrelevant in most cases, so in most cases you should consider merging the "1 to 1" tables into a single table.