According to Morgan Stanley’s Kathryn Huberty, Apple AAPL is the computer maker with the “most upside” as the PC market begins to stabilize after the dismal first quarter of 2009.

There’s some good news for Hewlett Packard HPQ and Dell DELL in the report to clients Huberty issued overnight Wednesday, but it’s mostly attributed to enterprise cycles and inventory restocking.

Apple, however, is a different story.

“Even before the new Macbook Pros launched,” she writes, “Apple began to outperform the broader commercial PC segment — with commercial Mac shipments up 25% [month over month] in May versus market growth of just 1%.”

The fact that the new laptops arrived in early June means that they will provide what Huberty calls “a catalyst for growth” in both the June and September fiscal quarters. She points to NPD weekly shipment data (reproduced in the chart at left) showing steady acceleration of Mac shipments over the past few weeks. “Lastly,” she concludes, “suppliers have recently noted Mac unit upside in the quarter.”

Huberty is raising her forecast for Mac sales in the second calendar quarter (Apple’s fiscal Q3) to 2.5 million units, up from 2.4 million. That would represent 12% quarter to quarter growth — less than Apple’s 14% average over the past three years, but a lot better than the 4% QtoQ decline last quarter.

For the fiscal quarter than ended Saturday, she expects Apple to report earnings of $1.16 a share on PC revenue (i.e., not including iPhones, iPods, etc.) of $3.072 billion, up a point or two from her previous estimates.

Huberty has not always been so bullish on the Mac. In fact, one of her reports last September helped trigger the sharpest one-day fall in Apple’s share price in eight years, one that wiped $18 billion off the company’s market cap in the space of 60 minutes. See Why Apple’s shares took a nosedive.