Spell Catcher X 10.2.1 and later are fully compatible with Mac OS X 10.4 (Tiger).
First things first: Make sure you are using Mac OS X 10.4.3 or later! A bug related to the input menu (in general, not at all specific to Spell Catcher) was fixed by Apple in Mac OS X 10.4.3.
In the version of Apple's Mail application that comes with Mac OS X 10.4, you have to turn off the Interactive preference to "Make replacements directly without backspacing". The same may also apply to Safari on Mac OS X 10.4. The new editing view used to compose mail messages and edit HTML (an editable WebView) does not fully support TSM Document Access, even though it claims to. Sigh.
Be aware that some of the default keyboard shortcuts have changed in Mac OS X 10.4. A couple of the default choices for Spell Catcher X's input menu conflict with the new Tiger defaults. One in particular is the Complete command in the input menu (command-F5). On Tiger, this is the default shortcut to turn VoiceOver on and off. We now use command-F6 starting in Spell Catcher X 10.2.1.
What About Earlier Versions of Spell Catcher X?
No - earlier versions of Spell Catcher X, including 10.1.3, should not be used on Tiger. The exact problem with earlier versions of Spell Catcher X is described in this FAQ. A number of other (perhaps old or otherwise outdated versions as well) service-providing applications have this exact same issue. See the May 4, 2005 issue of MacFixIt.
Upgrading to Tiger
If you perform an upgrade install, be aware that the Spell Catcher.component (input method, the item that appears in the Input Menu tab of International System Preferences, required for interactive checking) should be copied from your previous home folder, but will not be activated by default. You'll likely have to choose Spell Catcher from the input menu one time yourself. If, after upgrading to Tiger, you don't see Spell Catcher in the list of Input Menu items in International System Preferences, check to see that the Spell Catcher.component file exists in your home ~/Library/Components directory. If it doesn't exist, visit Spell Catcher's Interactive Preferences - there will be a button there you can click to install it. More about activating Spell Catcher's input method can be found in this FAQ.
But Wait - Cool New Things are Possible on Tiger!
Tiger is chock full of new, cool, incredibly useful features. While somewhat mundane compared to Spotlight, Dashboard, Automator and the like, there are a few new things that you'll appreciate when using Spell Catcher X on Mac OS X 10.4.
Vastly improved RTF capabilities
Take a look at the Format -> Text menu in TextEdit. Notice anything new? Tables and Lists are the big additions. The built-in OS X RTF translators can now handle reasonably complex tables and other similar objects. If you use Spell Catcher's Check Selection feature, your tables and lists will now be preserved when pasting your changes back into the document. While copying and pasting complex tables into and out of Word documents works quite well, it's not a perfect translation. Still, a whole lot better than it was pre-10.4!
New Input Source Options
Open System Preferences, click the International Icon, take a look at the Input Menu tab. The age-old "Font and Keyboard" synchronization options have been replaced with something far more useful (and understandable!). Specifically, the "Allow a different input source for each document" option means that Spell Catcher's input method can be active only in those applications (and more precisely, in those documents within an application) that you need it to be active in. Give it a try - open TextEdit, make two new untitled documents. Activate Spell Catcher's input method in one of them. Choose a keyboard layout in another. Switch between documents, watch the input menu.
The first radio button (Use one input source) will give you behavior similar to what you're probably used to with previous versions of Mac OS X (not exactly, but similar). The second (Allow a different input source) gives you more flexibility if you're someone that uses different keyboard layouts and/or input methods in your normal workflow. Try them both, see what works for you best.