CPU usage on RDS instance monotonically increasing with no change to query volume
I have some queries for you regarding table sizes that you can run in MySQL during these spikes

Database size in terms of StorageEngine (MB)

SELECT IFNULL(B.engine,'Total') "Storage Engine", CONCAT(LPAD(REPLACE(FORMAT(
B.DSize/POWER(1024,pw),3),',',''),17,' '),' ',SUBSTR(' KMGTP',pw+1,1),'B') "Data Size",
CONCAT(LPAD(REPLACE(FORMAT(B.ISize/POWER(1024,pw),3),',',''),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Index Size",CONCAT(LPAD(REPLACE(FORMAT(B.TSize/
POWER(1024,pw),3),',',''),17,' '),' ',SUBSTR(' KMGTP',pw+1,1),'B') "Table Size"
FROM (SELECT engine,SUM(data_length) DSize,
SUM(index_length) ISize,SUM(data_length+index_length) TSize FROM information_schema.tables
WHERE table_schema NOT IN ('mysql','information_schema','performance_schema') AND
engine IS NOT NULL GROUP BY engine WITH ROLLUP) B,(SELECT 2 pw) A ORDER BY TSize;


Database size in terms of Databases (MB)

SELECT DBName,CONCAT(LPAD(FORMAT(SDSize/POWER(1024,pw),3),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Data Size",
CONCAT(LPAD(FORMAT(SXSize/POWER(1024,pw),3),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Index Size",
CONCAT(LPAD(FORMAT(STSize/POWER(1024,pw),3),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Total Size" FROM (SELECT
IFNULL(DB,'All Databases') DBName,SUM(DSize) SDSize,SUM(XSize) SXSize,
SUM(TSize) STSize FROM (SELECT table_schema DB,data_length DSize,
index_length XSize,data_length+index_length TSize FROM information_schema.tables
WHERE table_schema NOT IN ('mysql','information_schema','performance_schema')) AAA
GROUP BY DB WITH ROLLUP) AA,(SELECT 2 pw) BB ORDER BY (SDSize+SXSize);


Database size in terms of Database/StorageEngine (MB)

SELECT IF(ISNULL(B.table_schema)+ISNULL(B.engine)=2,"Storage for All Databases",
IF(ISNULL(B.table_schema)+ISNULL(B.engine)=1,CONCAT("Storage for ",B.table_schema),
CONCAT(B.engine," Tables for ",B.table_schema))) Statistic,CONCAT(LPAD(REPLACE(FORMAT(
B.DSize/POWER(1024,pw),3),',',''),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Data Size",CONCAT(LPAD(REPLACE(FORMAT(
B.ISize/POWER(1024,pw),3),',',''),17,' '),' ',SUBSTR(' KMGTP',pw+1,1),'B') "Index Size",
CONCAT(LPAD(REPLACE(FORMAT(B.TSize/POWER(1024,pw),3),',',''),17,' '),' ',
SUBSTR(' KMGTP',pw+1,1),'B') "Table Size" FROM (SELECT table_schema,engine,
SUM(data_length) DSize,SUM(index_length) ISize,SUM(data_length+index_length) TSize
FROM information_schema.tables WHERE table_schema NOT IN
('mysql','information_schema','performance_schema') AND engine IS NOT NULL
GROUP BY table_schema,engine WITH ROLLUP) B,(SELECT 2 pw) A ORDER BY TSize;


Pay attention to certain markers


Innodb_buffer_pool_pages_dirty
Innodb_data_reads
Innodb_data_writes


I recommend downloading MySQL Administrator (I know, it's old but I still you it for quick and dirty "I WANNA SEE STATS NOW" moments of day) and set it up. I customized my own graphs to watch the size of the InnoDB Buffer Pool and its dirty pages. You could also just use the Connection Health tab.