The charts on this page will give you a good idea of what you can expect to start with in regard to doubles, count, trumps, etc.
It's often interesting after you see something "rare" to know just how rare it was.

Starting with different number of suits:	Probabilities
Hands with 7 suits	0.423140266 42%
Hands with 6 suits (void in only 1)	0.469823655 47%
Hands with 5 suits (void in 2)	0.103488902 10%
Hands with 4 suits (void in 3)	0.003547177 0.35% (1 in 282)

This is interesting because almost 90% of the time you will have either one of each suit, or be void in only one.
For fun, I have also counted hands that have no more than two trumps of any kind. I have given them names, as follows:
Bad hand (or Noah's Ark hand) - one that has a maximum trump count of any kind at two
You can expect to get one of those 1 out of every 35 hands, or 2.85% of the time.
Terrible hand - a "bad hand" but also with no doubles
You will get one of those only 1 out of every 2546 hands!
Horrible hand - a "terrible hand" but also with no count
Luckily those only appear 1 out of every 11,385 hands!

Number of trumps you will start with:	Probabilities	Notes
Probability of starting w/7 Trumps	0.000007	7 (1 in 148,005)
Probability of starting w/6 Trumps	0.000993	6 or more 0.001000 (1 in 1,000)
Probability of starting w/5 Trumps	0.029796	5 or more 0.030796 (roughly 1 in 32)
Probability of starting w/4 Trumps	0.305057	4 or more 0.335854 (roughly 1 in 3)
Probability of starting w/3 Trumps	0.635654	3 or more 0.971508 (97 out of 100)
Probability of starting w/2 Trumps	0.028492	2 or more 1.000000 (always)

So, you will almost always have three of something. But of course, that does not mean that you will be able to bid on them

Doubles you will start with:	Probabilities	Notes
starting w/ 7 doubles	0.000001	(1 in 1.184 million)
starting w/ 6 or more doubles	0.000125	(1 in 8K)
starting w/ 5 or more doubles	0.003850	(roughly 1 in 260)
starting w/ 4 or more doubles	0.043164	(roughly 1 in 23)
starting w/ 3 or more doubles	0.220080	(roughly 2 in 9)
starting w/ 2 or more doubles	0.580987	(roughly 4 in 7)
starting w/ 1 or more doubles	0.901794	(roughly 9 in 10)
starting w/ no doubles	0.098206	(roughly 1 in 10)

EXACT Doubles you will start with:	Probabilities	Notes
starting w/ exactly 7 doubles	0.000001	(1 in 1.184 million)
starting w/ exactly 6 doubles	0.000124	(1 in 8K)
starting w/ exactly 5 doubles	0.003725	(roughly 1 in 268)
starting w/ exactly 4 doubles	0.039315	(roughly 1 in 25)
starting w/ exactly 3 doubles	0.176915	(roughly 2 in 11)
starting w/ exactly 2 doubles	0.360908	(roughly 4 in 11)
starting w/ exactly 1 doubles	0.320807	(roughly 1 in 3)
starting w/ no doubles	0.098206	(roughly 1 in 10)

Team Doubles

Number of doubles your team will start with:	Probabilities	Notes
Odds of team starting w/7 doubles	0.002899	(1 in 345)
team w/6 or more doubles	0.038406	(1 in 26)
team w/5 or more doubles	0.192271	(roughly 1 in 5)
team w/4 or more doubles	0.500000	(1 in 2)
team w/3 or more doubles	0.807729	(roughly 4 in 5)
team w/2 or more doubles	0.961594
team w/1 or more doubles	0.997101
team w/no doubles	0.002899	(1 in 345)

EXACT Number of doubles your team will start with:	Probabilities	Notes
team w/exactly 7 doubles	0.002899	(1 in 345)
team w/exactly 6 doubles	0.035507	(1 in 28)
team w/exactly 5 doubles	0.153865	(2 in 13)
team w/exactly 4 doubles	0.307729	(4 in 13)
team w/exactly 3 doubles	0.307729	(4 in 13)
team w/exactly 2 doubles	0.153865	(2 in 13)
team w/exactly 1 double	0.035507	(1 in 28)
team w/exactly no doubles	0.002899	(1 in 345)

Count you will start with:	Probabilities	Notes
Hands w/ 0 count	0.207051	(roughly 1 in 5)
Hands w/ 5 count	0.255769	(roughly 1 in 4)
Hands w/ 10 count	0.255769	(roughly 1 in 4)
Hands w/ 15 count	0.177991	(roughly 2 in 11)
Hands w/ 20 count	0.073291	(roughly 2 in 27)
Hands w/ 25 count	0.025427	(roughly 1 in 39)
Hands w/ 30 count	0.0044872	(1 in 223)
Hands w/ 35 count	0.0002137	(1 in 4680)