What motif finding software is available for multiple sequences ~10Kb? I have around ~3,000 short sequences of approximately ~10Kb long. What are the best ways to find the motifs among all of these sequences? Is there a certain software/method recommended?

There are several ways to do this. My goal would be to:

(1) Check for motifs repeated within individual sequences

(2) Check for motifs shared among all sequences

(3) Check for the presence of "expected" or known motifs

With respect to #3, I'm also curious if I find e.g. trinucleotide sequences, how does one check the context around these regions?

Thank you for the recommendations/help!
The MEME Suite web site contains a collection of tools for motif analysis (I'm one of the maintainers). It contains two de novo motif discovery tools: MEME and DREME. Public web applications are provided, but you can also download and build command line tools for a local installation.

For your first goal you could use MEME and select the "Any number of repetitions model" (ANR). For your second goal, you'd use MEME with the "Zero or One Occurrences Per Sequence" (ZOOPS) model. For your third goal you could use FIMO (Find Individual Motif Occurrences), and one or more of the motif databases provided on the software and database download page.

It sounds like your sequence data is about 30Mb. The MEME web application is limited to 60kb of sequence data, so you'd have to install a local copy of the MEME Suite. MEME would take a long time to analyze a 30Mb sequence database unless you have MPI configured, and lots of cores available. You might want to consider analyzing a randomly selected subset of your sequences. The running time of MEME grows as the cube of the number of sequences. 

For short motifs, you may want to use DREME rather than MEME. DREME is better than MEME at identifying short motifs, but is limited to motifs &lt;= 8 positions wide.